singular

en-adj

singular (comparative more singular, superlative most singular)

more singular (comparative)
most singular (superlative)
sing. (alternative abbreviation)
sg. (alternative abbreviation)
sg (alternative abbreviation)

From Middle English singuler, from Old French, from Latin singulāris (“alone of its kind”), from Latin singulus (“single”).

(linear algebra, of matrix) Having no inverse.

особен

linear algebra: of matrix: having no inverse

(linear algebra, of transformation) Having the property that the matrix of coefficients of the new variables has a determinant equal to zero.


dual

PIE word
 *dwóh₁
Borrowed from Latin dualis (“two”), from duo (“two”) + adjective suffix -alis.

Accordingly, a hyperplane in the sample space is dual to a subspace in the variable space.

2012, Doug Fisher, Hans-J. Lenz, Learning from Data: Artificial Intelligence and Statistics V, Springer Science & Business Media, →ISBN, page 81:

(linear algebra) Being the space of all linear functionals of (some other space).


dimension

(linear algebra) The number of elements of any basis of a vector space.


bra

bras (plural)
brassiere (alternative dated)
brassière (alternative dated)

From bracket, referring to the notation introduced in 1939 by Paul Dirac.

(linear algebra, quantum mechanics) One of the two vectors in the standard notation for describing quantum states in quantum mechanics, the row vector; the other (column) vector being its complex conjugate, the ket.


base

From Latin basis, from Ancient Greek βάσις (básis).

Base on the Spanish Wikipedia.Wikipedia ᵉˢ


basis

en-noun

basis (plural bases or (rare) baseis or (nonstandard) basises)

bases (plural)
baseis (plural rare)
basises (nonstandard plural)

From Latin basis, from Ancient Greek βάσις (básis), from Proto-Indo-European *gʷémtis, derived from Proto-Indo-European *gʷem- (whence also come). Doublet of base.

linearly independent
linearly independent

(linear algebra) In a vector space, a linearly independent set of vectors spanning the whole vector space.

բազիս

linearly independent, spanning set of vectors


basis

From Latin basis, from Ancient Greek βᾰ́σῐς (bắsĭs). Doublet of base, and also related to komme, from the same ultimate source.


similarity

similarity (countable and uncountable, plural similarities)

From French similarité.
Morphologically similar + -ity

(mathematics, linear algebra) The property of two matrices being similar.


vapaa

+obj

[with illative of third infinitive ‘to do’ (with verbs)]

fi-adj

vapaa (comparative vapaampi, superlative vapain)

vapaampi (comparative)
vapain (superlative)
vapaa (nominative singular)
vapaat (nominative plural)
vapaa (accusative nominative singular)
vapaat (accusative nominative plural)
vapaan (accusative genitive singular)
vapaat (accusative genitive plural)
vapaan (genitive singular)
vapaiden (genitive plural)
vapaitten (genitive plural)
vapaata (partitive singular)
vapaita (partitive plural)
vapaassa (inessive singular)
vapaissa (inessive plural)
vapaasta (elative singular)
vapaista (elative plural)
vapaaseen (illative singular)
vapaisiin (illative plural)
vapaihin (illative plural rare)
vapaalla (adessive singular)
vapailla (adessive plural)
vapaalta (ablative singular)
vapailta (ablative plural)
vapaalle (allative singular)
vapaille (allative plural)
vapaana (essive singular)
vapaina (essive plural)
vapaaksi (singular translative)
vapaiksi (plural translative)
vapaatta (abessive singular)
vapaitta (abessive plural)
vapain (instructive plural)
vapaine (comitative plural)
vapaani (nominative singular)
vapaani (nominative plural)
vapaani (accusative nominative singular)
vapaani (accusative nominative plural)
vapaani (accusative genitive singular)
vapaani (accusative genitive plural)
vapaani (genitive singular)
vapaideni (genitive plural)
vapaitteni (genitive plural)
vapaatani (partitive singular)
vapaitani (partitive plural)
vapaassani (inessive singular)
vapaissani (inessive plural)
vapaastani (elative singular)
vapaistani (elative plural)
vapaaseeni (illative singular)
vapaisiini (illative plural)
vapaihini (illative plural rare)
vapaallani (adessive singular)
vapaillani (adessive plural)
vapaaltani (ablative singular)
vapailtani (ablative plural)
vapaalleni (allative singular)
vapailleni (allative plural)
vapaanani (essive singular)
vapainani (essive plural)
vapaakseni (singular translative)
vapaikseni (plural translative)
vapaattani (abessive singular)
vapaittani (abessive plural)
vapaineni (comitative plural)
vapaasi (nominative singular)
vapaasi (nominative plural)
vapaasi (accusative nominative singular)
vapaasi (accusative nominative plural)
vapaasi (accusative genitive singular)
vapaasi (accusative genitive plural)
vapaasi (genitive singular)
vapaidesi (genitive plural)
vapaittesi (genitive plural)
vapaatasi (partitive singular)
vapaitasi (partitive plural)
vapaassasi (inessive singular)
vapaissasi (inessive plural)
vapaastasi (elative singular)
vapaistasi (elative plural)
vapaaseesi (illative singular)
vapaisiisi (illative plural)
vapaihisi (illative plural rare)
vapaallasi (adessive singular)
vapaillasi (adessive plural)
vapaaltasi (ablative singular)
vapailtasi (ablative plural)
vapaallesi (allative singular)
vapaillesi (allative plural)
vapaanasi (essive singular)
vapainasi (essive plural)
vapaaksesi (singular translative)
vapaiksesi (plural translative)
vapaattasi (abessive singular)
vapaittasi (abessive plural)
vapainesi (comitative plural)
vapaamme (nominative singular)
vapaamme (nominative plural)
vapaamme (accusative nominative singular)
vapaamme (accusative nominative plural)
vapaamme (accusative genitive singular)
vapaamme (accusative genitive plural)
vapaamme (genitive singular)
vapaidemme (genitive plural)
vapaittemme (genitive plural)
vapaatamme (partitive singular)
vapaitamme (partitive plural)
vapaassamme (inessive singular)
vapaissamme (inessive plural)
vapaastamme (elative singular)
vapaistamme (elative plural)
vapaaseemme (illative singular)
vapaisiimme (illative plural)
vapaihimme (illative plural rare)
vapaallamme (adessive singular)
vapaillamme (adessive plural)
vapaaltamme (ablative singular)
vapailtamme (ablative plural)
vapaallemme (allative singular)
vapaillemme (allative plural)
vapaanamme (essive singular)
vapainamme (essive plural)
vapaaksemme (singular translative)
vapaiksemme (plural translative)
vapaattamme (abessive singular)
vapaittamme (abessive plural)
vapainemme (comitative plural)
vapaanne (nominative singular)
vapaanne (nominative plural)
vapaanne (accusative nominative singular)
vapaanne (accusative nominative plural)
vapaanne (accusative genitive singular)
vapaanne (accusative genitive plural)
vapaanne (genitive singular)
vapaidenne (genitive plural)
vapaittenne (genitive plural)
vapaatanne (partitive singular)
vapaitanne (partitive plural)
vapaassanne (inessive singular)
vapaissanne (inessive plural)
vapaastanne (elative singular)
vapaistanne (elative plural)
vapaaseenne (illative singular)
vapaisiinne (illative plural)
vapaihinne (illative plural rare)
vapaallanne (adessive singular)
vapaillanne (adessive plural)
vapaaltanne (ablative singular)
vapailtanne (ablative plural)
vapaallenne (allative singular)
vapaillenne (allative plural)
vapaananne (essive singular)
vapainanne (essive plural)
vapaaksenne (singular translative)
vapaiksenne (plural translative)
vapaattanne (abessive singular)
vapaittanne (abessive plural)
vapainenne (comitative plural)
vapaansa (nominative singular)
vapaansa (nominative plural)
vapaansa (accusative nominative singular)
vapaansa (accusative nominative plural)
vapaansa (accusative genitive singular)
vapaansa (accusative genitive plural)
vapaansa (genitive singular)
vapaidensa (genitive plural)
vapaittensa (genitive plural)
vapaataan (partitive singular)
vapaatansa (partitive singular)
vapaitaan (partitive plural)
vapaitansa (partitive plural)
vapaassaan (inessive singular)
vapaassansa (inessive singular)
vapaissaan (inessive plural)
vapaissansa (inessive plural)
vapaastaan (elative singular)
vapaastansa (elative singular)
vapaistaan (elative plural)
vapaistansa (elative plural)
vapaaseensa (illative singular)
vapaisiinsa (illative plural)
vapaihinsa (illative plural rare)
vapaallaan (adessive singular)
vapaallansa (adessive singular)
vapaillaan (adessive plural)
vapaillansa (adessive plural)
vapaaltaan (ablative singular)
vapaaltansa (ablative singular)
vapailtaan (ablative plural)
vapailtansa (ablative plural)
vapaalleen (allative singular)
vapaallensa (allative singular)
vapailleen (allative plural)
vapaillensa (allative plural)
vapaanaan (essive singular)
vapaanansa (essive singular)
vapainaan (essive plural)
vapainansa (essive plural)
vapaakseen (singular translative)
vapaaksensa (singular translative)
vapaikseen (plural translative)
vapaiksensa (plural translative)
vapaattaan (abessive singular)
vapaattansa (abessive singular)
vapaittaan (abessive plural)
vapaittansa (abessive plural)
vapaineen (comitative plural)
vapainensa (comitative plural)

From Proto-Finnic *vapada, probably borrowed from early Proto-Slavic *svabadā (later *svoboda (“freedom”)).

(linear algebra) linearly independent (of a set of vectors)


基

Characters in the same phonetic series (其) (Zhengzhang, 2003)
Phono-semantic compound (形聲 /形声, OC *kɯ): phonetic 其 (OC *kɯ, *ɡɯ) + semantic 土 (“earth”) – a foundation of earth.

Han compound

𡉊
𡉨
𡉝
𡋵
踑
𨄎
㚻
gay

(linear algebra) basis (Classifier: 組／组 m; 個／个 m)


核

Phono-semantic compound (形聲 /形声, OC *ɡuːd, *ɡrɯːɡ): semantic 木 (“tree”) + phonetic 亥 (OC *ɡɯːʔ) – a kind of tree. Modern meanings are borrowed.
The definitions related to “to examine” come from the “kernel” sense, i.e. “to go to the core” (Schuessler, 2007).
Generally thought to be of Sino-Tibetan origin. It has been compared to Tibetan རག་ཙེ (rag tse), Karbi rak (Gong, 2002; Schuessler, 2007). However, Hill (2012) rejects the connection to the Tibetan word because most words ending with ཙེ (tse) are loanwords from Chinese words ending with the 子 suffix.
Smith (2011) groups this word with the following terms into a word family meaning “root, germ, generative core”:
* 荄 (OC *kɯː, “grassroot”) and its derivative 根 (OC *kɯːn, “root”),
* 骸 (OC *ɡrɯː, “(human) bones”) (also noted by Schuessler, 2007),
* 孩 (OC *ɡɯː, “child”),
* 亥 (OC *ɡɯːʔ, “sprouting stage”), i.e. "the moon’s first appearance" > "twelfth earthly branch".

Han compound

(mathematics, algebra, linear algebra) kernel


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

ядро

(mathematics, algebra) set of elements mapped to zero

algebraic structures
algebraic structure

linear transformation
linear transformation

(mathematics, algebra, loosely, of a morphism f between algebraic structures) The set of elements of the domain of f which are mapped to an identity element.

(mathematics, linear algebra, of a linear transformation f:M→N between modules or vector spaces) The set of elements of M which are mapped to the additive identity 0 of N.


syzygy

en-noun

syzygy (countable and uncountable, plural syzygies)

From Late Latin sȳzygia (“conjunction”), from Ancient Greek συζυγία (suzugía, “union, yoke, pair”). This word was recognized as English in 1847 (in its astronomical meaning).

der

A finitely presented module is an A-module M given by a finite number of generators and relations. Therefore it is a module with a finite generator set having a finitely generated syzygy module.

2015, Henri Lombardi, Claude Quitté, Commutative Algebra: Constructive Methods: Finite Projective Modules, Springer, →ISBN, page 173:

(mathematics, linear algebra) A relation between generators of a module.


譜

Phono-semantic compound (形聲 /形声, OC *paːʔ): semantic 言 (“sound”) + phonetic 普 (OC *pʰaːʔ).

Han compound

(linear algebra) spectrum (the set of eigenvalues of a matrix)


spectrum

From Latin spectrum (“appearance, image, apparition”), from speciō (“look at, view”). Doublet of specter. See also scope.

bor

(mathematics, linear algebra) The set of eigenvalues of a matrix.


homogeneous

From Medieval Latin homogeneus, from Ancient Greek ὁμογενής (homogenḗs, “of the same race, family or kind”), from ὁμός (homós, “same”) + γένος (génos, “kind”). Compare homo- (“same”) and -ous (adjectival suffix).

der

(mathematics) In any of several technical senses uniform; scalable; having its behavior or form determined by, or the same as, its behavior on or form at a smaller component (of its domain of definition, of itself, etc.).

Of polynomials, functions, equations, systems of equations, or linear maps:

(linear algebra, by specialization, of a system of linear equations) Such that all the constant terms are zero.


scalar product

Coined by William Kingdon Clifford in 1877 in his book Elements of Dynamic.

coin

(linear algebra) The product of two vectors computed as the sum of the corresponding elements of the vectors, or, equivalently, as the product of the magnitudes of the vectors and the cosine of the angle between their directions.


vector product

(linear algebra) a vector with the size given by the product of two vectors computed as the product of the magnitudes of the vectors and the sine of the angle between their directions, and directed perpendicular to the given two vectors, with positive orientation.

productu vectorial

vector with size given by the product of two vectors


cross product

LL-Q1860 (eng)-Flame, not lame-cross product.wav

Coined by American scientist Josiah Willard Gibbs in 1881.

The cross product can be derived like so: given a pair of vectors #92;xi and #92;eta, solve for a vector #92;zeta which is perpendicular to both #92;xi and #92;eta, i.e., whose dot product with both #92;xi and #92;eta is zero.


dot product

In reference to the symbol · used to denote the operation.


flag

en-noun

flag (countable and uncountable, plural flags)

From Middle English flag, flagge (“flag”), further etymology uncertain. Perhaps from or related to early Middle English flage (name for a baby's garment) and Old English flagg, flacg (“cataplasm, poultice, plaster”). Or, perhaps ultimately imitative, or otherwise drawn from Proto-Germanic *flaką (“something flat”), from Proto-Indo-European *pleh₂- (“flat, broad, plain”), referring to the shape.
Germanic cognates include Saterland Frisian Flaage (“flag”), West Frisian flagge (“flag”), Dutch vlag (“flag”), German Flagge (“flag”), Swedish flagga (“flag”), Danish flag (“flag, ship's flag”). Compare also Middle English flacken (“to flutter, palpitate”), Swedish dialectal flage (“to flutter in the wind”), Old Norse flögra (“to flap about”). Akin to Old High German flogarōn (“to flutter”), Old High German flogezen (“to flutter, flicker”), Middle English flakeren (“to move quickly to and fro”), Old English flacor (“fluttering, flying”). More at flack, flacker.

onomatopoeic

imitative

cog

(mathematics, linear algebra) A sequence of subspaces of a vector space, beginning with the null space and ending with the vector space itself, such that each member of the sequence (until the last) is a proper subspace of the next.

lippu

linear algebra: sequence of subspaces of a vector space


similar

en-adj

similar (comparative more similar, superlative most similar)

more similar (comparative)
most similar (superlative)

From French similaire, from Medieval Latin similaris, extended from Latin similis (“like”); akin to simul (“together”).

(mathematics, linear algebra) Of two square matrices; being such that a conjugation sends one matrix to the other.


auto-

Borrowed from Ancient Greek αὐτός (autós, “self”).

bor+

(Brazil, linear algebra) eigen- (relating to eigenvectors and eigenvalues)


eigen

cy-adj

eigen (feminine singular eigen, plural eigen, not comparable)

eigen (feminine singular)
eigen (plural)

(linear algebra) eigen-, Forms terms pertaining to or related to mathematical objects uaffected by a given linear transformation, except by scalar multiplication.


eigen-

Borrowed from German eigen (“own”). Many of the compounds represent partial translations from German, e.g. eigenvalue from German Eigenwert. Doublet of own.

(linear algebra) Forms terms pertaining to or related to eigenvectors, eigenvalues; especially for naming mathematical objects which are not affected by a given linear transformation except for by scalar multiplication.


congruence

en-noun

congruence (countable and uncountable, plural congruences)

/ˈkɒŋ.ɡɹʊ.əns/

/ˈkɔŋ.ɡɹʉ.əns/

From Middle English congruence, from Latin congruentia (“agreement”), from congruēns, present active participle of congruō (“meet together, agree”).

համաձայնություն

the quality of agreeing or corresponding

(mathematics, linear algebra) Matrix similarity by an orthogonal matrix.

kongruenssi

linear algebra, matrix similarity by an orthogonal matrix


unitary

confederal

antonym(s) of “that concentrates power in a single body”

unitary group

group whose elements are unitary matrices or operators

The eigenvectors of an orthogonal or unitary operator, corresponding to different eigenvalues, are orthogonal.

1997, P. K. Suetin, Alexandra I. Kostrikin, Yuri I. Manin, translated by M. E. Alferieff, Linear Algebra and Geometry, page 137:

In essence we are looking for some way to average the individual unitary matrices Uₖ. But a linear combination of unitary matrices does not remain unitary.

2002, M. Klajman, J. A. Chambers, “A Novel Approximate Joint Diagonalization Algorithm”, in J. G. McWhirter, I. K. Proudler, editors, Mathematics in Signal Processing V, page 71:

We then repeat the same procedure to the (d − 1) × (d − 1) block unitary matrix using (d − 2) two-level unitary matrices.

2008, Mikio Nakahara, Tetsuo Ohmi, Quantum Computing: From Linear Algebra to Physical Realizations, page 84:

mathematical analysis
mathematical analysis

(mathematics, linear algebra, mathematical analysis, of a matrix or operator) Whose inverse is equal to its adjoint.

ортогонален

mathematics: that has an inverse equal to its adjoint


determinant

Borrowed from French déterminant, equivalent to determine + -ant.

1966 [Allyn & Bacon], Howard Whitley Eves, Elementary Matrix Theory, 1980, Dover, Unabridged corrected republication, page 165,
The determinant of a square matrix A_(n) is a function (actually a polynomial function) of the elements a_ij of A.

1990, Assem S. Deif, Advanced Matrix Theory for Scientists and Engineers, Gordon and Breach Science Publishers (Abacus Press), 2nd Edition, page 18,
Show that the determinant of a Hermitian matrix is real and that of a skew-Hermitian matrix is imaginary.

Using this definition, we derive the basic properties of a determinant that are useful in its evaluation. In particular, it is shown how the calculation of a determinant can be reduced to the calculation of determinants of lower order.

2009, Richard A. Brualdi, Dragoš Cvetkoviċ, A Combinatorial Approach to Matrix Theory and Its Applications, Taylor & Francis (CRC / Chapman & Hall), page 63:

The following proposition shows a connection between negafibonacci numbers and the determinants of a family of tridiagonal matrices. […] Assuming that the determinant det(Gₖ) = f₋₍ₖ₊₁₎ for all k ≤ n, we aim to show that det(Gₙ₊₂) = f₋₍ₙ₊₃₎.

2019 June 24, Juan Triana, “Negafibonacci Numbers via Matrices”, in Bulletin of TICMI, volume 23, number 1, →ISSN, archived from the original on 2019-07-13, Negafibonacci numbers as tridiagonal matrix determinants, pages 3, 4:

transformation matrix
transformation matrix

(linear algebra) A scalar that encodes certain characteristics of a given transformation matrix; the unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value 1 for the unit matrix; abbreviated as: det.

детерминанта

scalar that encodes certain characteristics of a square matrix


determinant

determinant (nominative singular)
determinanty (nominative plural)
determinantu (genitive singular)
determinantů (genitive plural)
determinantu (dative singular)
determinantům (dative plural)
determinant (accusative singular)
determinanty (accusative plural)
determinante (singular vocative)
determinanty (plural vocative)
determinantu (locative singular)
determinantech (locative plural)
determinantem (instrumental singular)
determinanty (instrumental plural)

determinant (unique scalar function over square matrices)


determinant

head

determinant c (singular definite determinanten, plural indefinite determinanter)

determinanten (definite singular)
determinanter (indefinite plural)
determinant (indefinite nominative singular)
determinanten (definite nominative singular)
determinanter (indefinite nominative plural)
determinanterne (definite nominative plural)
determinants (genitive indefinite singular)
determinantens (definite genitive singular)
determinanters (genitive indefinite plural)
determinanternes (definite genitive plural)


eigenvalue

From eigen- + value, a partial calque of German Eigenwert.

The eigenvalues #92;lambda of a square transformation matrix #92;mathrm#123;M#125; may be found by solving #92;det(#92;mathrm#123;M#125;-#92;lambda#92;mathrm#123;I#125;)#61;0.

In the extension, one associates eigenvalues, sets of scalars, with arrays of matrices by considering the singularity of linear combinations of the matrices in the various rows, involving the same coefficients in each case. Attention to this area was called in the early 1920's by R. D. Carmichael, who pointed out in addition the enormous variety of mixed eigenvalue problems with several parameters.

1972, F. V. Atkinson, Multiparameter Eigenvalue Problems, Volume I: Matrices and Compact Operators, Academic Press, page x:

For many quantum-mechanical problems it is important to investigate the change of eigenvalues and eigenfunctions with the continuous change of one or more parameters. The case in which one knows the eigenvalues and eigenfunctions for two special values of the parameters, and is interested in the region in between is particularly interesting.

2000, John von Neumann, E. Wigner, “On the Behaviour of Eigenvalues in Adiabatic Processes”, in Hinne Hettema, transl., edited by Hinne Hettema, Quantum Chemistry: Classic Scientific Papers, World Scientific, published 1929, page 25:

Problems that require an investigation of eigenvalues and eigenfunctions arise in connection with numerous topics in mechanics, the theory of vibrations and stability, hydrodynamics, elasticity, acoustics, electrodynamics, quantum mechanics, etc.

2005, Leonid D. Akulenko, Sergei V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications, CRC Press (Chapman & Hall), page 1:

To find all the eigenvalues of A, solve the characteristic equation.
For each eigenvalue λ, to find the corresponding set of eigenvectors, solve the linear system of equations […]

(Can we date this quote?), “Lesson 12”, in University College London, page 1:

(linear algebra) A scalar, λ, such that there exists a non-zero vector x (a corresponding eigenvector) for which the image of x under a given linear operator A is equal to the image of x under multiplication by λ; i.e. Ax=λx.


eigenvector

From eigen- + vector, a partial calque of German Eigenvektor.
The prefix eigen- (also used in eigenvalue) was first used in 1904, by David Hilbert, and was possibly inspired by a related usage by Hermann von Helmholtz.

Both equations give the relation x₁ = −x₂. Therefore the set of eigenvectors corresponding to λ = 2 is given by: […] An eigenvector corresponding to λ₁ = 2 is ( −1 1 ).

(Can we date this quote?), “Lesson 11”, in University College London, 8.2 Method to find eigenvalues and eigenvectors, page 1:

(linear algebra) A vector that is only scaled (not rotated out of its span) under a particular linear transformation; a left or right eigenvector depending on context; (more formally) given a linear transformation A, a vector x such that Ax=λx [or xA=λx] for some scalar λ (called the eigenvalue).

vector propiu

vector not rotated by linear transformation


eigenvector

From eigen + vector, calque of German Eigenvektor.


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(linear algebra, of a matrix) Which commutes with its conjugate transpose.


rank

en-noun

rank (countable and uncountable, plural ranks)

From Middle English rank (“line, row”), from Old French ranc, rang, reng (“line, row, rank”) (Modern French rang), from Frankish *hring (“ring”), from Proto-Germanic *hringaz (“something bent or curved”).
Akin to Old High German (h)ring, Old Frisian hring, Old English hring, hrincg (“ring”), Old Norse hringr (“ring, circle, queue, sword; ship”). Doublet of ring and rink.

cog

(linear algebra) The maximal number of linearly independent columns (or rows) of a matrix.


Use as a multiplication sign is usually attributed to English mathematician William Oughtred.

(1,2,5) × (3,4,−1) = (−22, 16, −2)

(linear algebra) Vector product, cross product.


(linear algebra, functional analysis) Dual space.


tensor

en-noun

tensor (plural tensors or (muscle) tensores)

Borrowed from New Latin tensor (“that which stretches”), equivalent to tense + -or. Anatomical sense from 1704.
Introduced in the 1840s by William Rowan Hamilton as an algebraic quantity unrelated to the modern notion of tensor.
The contemporary mathematical meaning was introduced (as German Tensor) by Woldemar Voigt (1898) and adopted in English from 1915 (in the context of general relativity), obscuring the earlier Hamiltonian sense. The mathematical object is so named because an early application of tensors was the study of materials stretching under tension. (See, for example, Cauchy stress tensor on Wikipedia.Wikipedia)

pedia

The tensor #92;alpha#95;#123;ij#125; should really be called a “tensor of second rank,” because it has two indexes. A vector—with one index—is a tensor of the first rank, and a scalar—with no index—is a tensor of zero rank.

1963, Richard Feynman, “Chapter 31, Tensors”, in The Feynman Lectures on Physics, volume II:

multidimensional array
multidimensional array

(mathematics, linear algebra, physics) A mathematical object that describes linear relations on scalars, vectors, matrices and other algebraic objects, and is represented as a multidimensional array.

(engineering, computing) A multidimensional array with (at least) two dimensions.


tensor

Ultimately or directly from Latin tensor.


permanent

From Middle English permanent, permanente, from Middle French permanent, from Latin permanēns, from permaneō (“I stay through”). First attested in the 15th century.

(linear algebra, combinatorics) Given an n×n matrix a_ij,, the sum over all permutations π, of ∏ᵢ₌₁ⁿa_iπ(i).


entry

en-noun

entry (countable and uncountable, plural entries)

exit

antonym(s) of “doorway that provides a means of entering a building”

Inherited from Middle English entre, from Old French entree (feminine past participle of the verb entrer, Modern French entrée). From Latin intrō. Doublet of entrada and entrée.

The entry in the second row and first column of this matrix is 6.

en:algebraic term at any position in a matrix

(linear algebra) A term at any position in a matrix.


zero vector

The zero vector of a given vector space is the space's additive identity.

(mathematics, linear algebra) A vector 0 in a vector space V such that for any v∈V, 0+v=v; a vector whose value in every dimension is zero; i.e., ⃑0=<0,0,…,0>.


singular value decomposition

singular value decomposition (plural singular value decompositions)

(linear algebra) A particular type of factorisation of a matrix into a product of three matrices, of which the second is a diagonal matrix that has as the entries on its diagonal the singular values of the original matrix.

奇异值分解

linear algebra: type of factorisation of a matrix


vector space

LL-Q1860 (eng)-Flame, not lame-vector space.wav

A vector space is a set of vectors which can be linearly combined.

Each vector space has a basis and dimension.

scalar multiplication
scalar multiplication

(algebra, geometry, topology) A set of elements called vectors, together with some field and operations called addition (mapping two vectors to a vector) and scalar multiplication (mapping a vector and an element in the field to a vector), satisfying a list of constraints; equivalently, a module over a field.


transpose

en-verb

transpose (third-person singular simple present transposes, present participle transposing, simple past and past participle transposed)

transposes (present singular third-person)
transposing (participle present)
transposed (participle past)
transposed (past)

From Middle English transposen, from Old French transposer, from Latin trānspositus, perfect passive participle of trānspōnō (“to put across”), from trāns (“across”) + pōnō (“to put”).

(transitive, linear algebra) To rearrange elements in a matrix, by interchanging their respective row and column positional indicators.

transponovat

mathematics: to rearrange elements in a matrix


transpose

(linear algebra) Created by transposing a specified matrix.


transpose

(linear algebra) The matrix derived from performing a transpose operation on a given matrix.


transpose

(linear algebra) The process of rearranging elements in a matrix, by interchanging their respective row and column positional indicators.


orthonormal

LL-Q1860 (eng)-Flame, not lame-orthonormal.wav

Blend of orthogonal + normal, equivalent to ortho- + normal.

Given a set of linearly independent vectors, it is often useful to convert them into an orthonormal set of vectors.

(Can we date this quote?), “Lesson 10”, in University College London, 6.4 Gram-Schmidt Process, page 2:

(functional analysis, geometry, linear algebra) Of a set of vectors, both orthogonal and normalized.

(functional analysis, geometry, linear algebra) Of a linear transformation: that preserves both angles and lengths.


projection

en-noun

projection (countable and uncountable, plural projections)

/pɹəˈd͡ʒɛkʃən/

From either the Middle French projection or its etymon, the Classical Latin prōiectiō (stem: prōiectiōn-), from prōiciō, equivalent to project + -ion. Compare the Modern French projection, the German Projektion, and the Italian proiezione.

(linear algebra) An idempotent linear transformation which maps vectors from a vector space onto a subspace.

projektio

linear algebra: idempotent linear transformation from a vector space onto a subspace


projection

(linear algebra, mathematics) projection


I

From the Etruscan letter 𐌉 (i), from the Ancient Greek letter Ι (I, iota), derived from the Phoenician letter 𐤉 (y, yod), from the Egyptian hieroglyph 𓂝.

(mathematics, linear algebra) identity matrix


O

From the Etruscan letter 𐌏 (o, “o”), from the Ancient Greek letter Ο (O, “omikron”), derived from the Phoenician letter 𐤏 (ʿ, “ʿayin”), from the Egyptian hieroglyph 𓁹.

(linear algebra, group theory) orthogonal group


echelon

Borrowed from French échelon (“rung; echelon”), from échelle (“ladder”) + -on (diminutive suffix). Échelle is derived from Latin scāla (“ladder”), from scandō (“to ascend, climb”), from Proto-Indo-European *skend- (“to jump”).

An echelon matrix is a matrix, not necessarily square, with the following two properties: (i) There is at least one non-zero entry; rows consisting entirely of zeros, if any, lie below rows with at least one non-zero entry. (ii) In each non-zero row after the first, the left-most non-zero entry lies to the right of the left-most non-zero entry in the preceding row. […] In each of the non-zero rows of an echelon matrix, the left-most non-zero entry is called the pivot, […]

2001, Malcolm Pemberton, Nicholas Rau, “Systems of Linear Equations”, in Mathematics for Economists: An Introductory Textbook, Manchester, New York, N.Y.: Manchester University Press, →ISBN, section 12.1 (Echelon Matrices), page 204:

(linear algebra) Of a matrix: having undergone Gaussian elimination with the result that the leading coefficient or pivot (that is, the first nonzero number from the left) of a nonzero row is to the right of the pivot of the row above it, giving rise to a stepped appearance in the matrix.


square matrix

square matrix (plural square matrixes or square matrices)

square matrixes (plural)
square matrices (plural)

(linear algebra) A matrix having the same number of rows as columns.


diag

(linear algebra) a function that takes an n-element vector a and outputs an n by n diagonal matrix A where A_ii=a_i


identity matrix

identity matrix (plural identity matrices or identity matrixes)

identity matrices (plural)
identity matrixes (plural)

(linear algebra) A diagonal matrix all of whose diagonal elements are equal to 1.

matriu identitat

diagonal matrix with all diagonal elements equal to 1


unit matrix

unit matrix (plural unit matrices or unit matrixes)

unit matrices (plural)
unit matrixes (plural)

(linear algebra) Synonym of identity matrix.


diagonal matrix

diagonal matrix (plural diagonal matrices)

(linear algebra) A matrix in which the entries outside the main diagonal are all zero.


main diagonal

In a diagonal matrix, non-zero elements appear only in the main diagonal.

(linear algebra) The entries of a square matrix from the top left to the bottom right.


diagonal element

diagonal element (plural diagonal elements)

(linear algebra) An element on the main diagonal of a square matrix; that is, an element in row k and column k, where k is an integer between 1 and the number of rows (or columns) in the matrix.


leading diagonal

leading diagonal (plural leading diagonals)


stochastic matrix

stochastic matrix (plural stochastic matrices)

(linear algebra, probability theory, statistics) A matrix having the property that the entries in each column are non-negative, real and sum to 1.


inverse matrix

inverse matrix (plural inverse matrices or inverse matrixes)

inverse matrices (plural)
inverse matrixes (plural)

Given the basis of some vector space V, how to find its dual basis, i.e., the basis of the dual space V#42;? Fill the columns of a square matrix M with the basis vectors of V. Find the inverse matrix M#123;-1#125; of M. Then the rows of M#123;-1#125; are the (co)vectors of that dual basis. Since (M#123;-1#125;)#123;-1#125;#61;M, then (V#42;)#42;#61;V.

(linear algebra) Of a matrix A, another matrix B such that A multiplied by B and B multiplied by A both equal the identity matrix.


self-dual

(linear algebra) Of a vector space, isomorphic to its dual space.

itseduaalinen

of a vector space, isomorphic to its dual space


A′ = Aᵀ


cofactor

(mathematics, linear algebra, of a matrix) The signed determinant of the submatrix produced by removing the row and column containing a specified element; primarily used in the recursive definition and calculation of the determinant of a matrix.


entrada

[en̪ˈt̪ɾa.ð̞ɐ]


entrada

[ẽn̪ˈt̪ɾa.ð̞a]


λ

Both equations give the relation x₁ = −x₂. Therefore the set of eigenvectors corresponding to λ = 2 is given by:

(Can we date this quote?), “Lesson 11”, in University College London (in English), 8.2 Method to find eigenvalues and eigenvectors, page 1:


affine transformation

affine transformation (plural affine transformations)

An affine transformation does not in general preserve angles between lines or distances between points, but it does preserve ratios of distances between points lying on a straight line.

Given an affine space X, every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X.

Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear and compositions of them in any combination and sequence.

Just as for plane transformations, we may show that the set of all affine transformations of space form a group.
Under an affine transformation of space, the image of a line is a line, and the image of a plane is a plane.

1965, P. S. Modenov, A. S. Parkhomenko, translated by Michael B. P. Slater, Geometric Transformations, Volume 1: Euclidean and Affine Transformations, Academic Press, page 145:

Theorem 15.2 A transformation such that the images of every three collinear points are themselves collinear is an affine transformation.
Are the affine transformations the same as those transformations for which the images of any three noncollinear points are themselves noncollinear? We shall see the answer is "Yes."

1982, George E. Martin, Transformation Geometry, Springer, page 169:

2004, Solomon Khmelnik, Computer Arithmetic of Geometrical Figures: Algorithms and Hardware Design, Mathematics in Computer Comp., page 8,
Most striking and well-known examples of affine transformation applications are computer tomography (see for instance [1]) and information compression for telecommunication systems (see [2]).
This book describes affine transformations (displacements, turns, scaling, shifts) of n-dimensional figures, where n=1,2,3,4.

(geometry, linear algebra) A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of an affine space: a mapping of an affine space onto itself that preserves both the dimension of any affine subspace and the ratio of the lengths of any pair of parallel line segments.

affine homomorphism

mappings from one space to a (generally) different one


kanta

kanta (nominative singular)
kannat (nominative plural)
kanta (accusative nominative singular)
kannat (accusative nominative plural)
kannan (accusative genitive singular)
kannat (accusative genitive plural)
kannan (genitive singular)
kantojen (genitive plural)
kantain (genitive plural rare)
kantaa (partitive singular)
kantoja (partitive plural)
kannassa (inessive singular)
kannoissa (inessive plural)
kannasta (elative singular)
kannoista (elative plural)
kantaan (illative singular)
kantoihin (illative plural)
kannalla (adessive singular)
kannoilla (adessive plural)
kannalta (ablative singular)
kannoilta (ablative plural)
kannalle (allative singular)
kannoille (allative plural)
kantana (essive singular)
kantoina (essive plural)
kannaksi (singular translative)
kannoiksi (plural translative)
kannatta (abessive singular)
kannoitta (abessive plural)
kannoin (instructive plural)
See the possessive forms below. (comitative)
kantani (nominative singular)
kantani (nominative plural)
kantani (accusative nominative singular)
kantani (accusative nominative plural)
kantani (accusative genitive singular)
kantani (accusative genitive plural)
kantani (genitive singular)
kantojeni (genitive plural)
kantaini (genitive plural rare)
kantaani (partitive singular)
kantojani (partitive plural)
kannassani (inessive singular)
kannoissani (inessive plural)
kannastani (elative singular)
kannoistani (elative plural)
kantaani (illative singular)
kantoihini (illative plural)
kannallani (adessive singular)
kannoillani (adessive plural)
kannaltani (ablative singular)
kannoiltani (ablative plural)
kannalleni (allative singular)
kannoilleni (allative plural)
kantanani (essive singular)
kantoinani (essive plural)
kannakseni (singular translative)
kannoikseni (plural translative)
kannattani (abessive singular)
kannoittani (abessive plural)
kantoineni (comitative plural)
kantasi (nominative singular)
kantasi (nominative plural)
kantasi (accusative nominative singular)
kantasi (accusative nominative plural)
kantasi (accusative genitive singular)
kantasi (accusative genitive plural)
kantasi (genitive singular)
kantojesi (genitive plural)
kantaisi (genitive plural rare)
kantaasi (partitive singular)
kantojasi (partitive plural)
kannassasi (inessive singular)
kannoissasi (inessive plural)
kannastasi (elative singular)
kannoistasi (elative plural)
kantaasi (illative singular)
kantoihisi (illative plural)
kannallasi (adessive singular)
kannoillasi (adessive plural)
kannaltasi (ablative singular)
kannoiltasi (ablative plural)
kannallesi (allative singular)
kannoillesi (allative plural)
kantanasi (essive singular)
kantoinasi (essive plural)
kannaksesi (singular translative)
kannoiksesi (plural translative)
kannattasi (abessive singular)
kannoittasi (abessive plural)
kantoinesi (comitative plural)
kantamme (nominative singular)
kantamme (nominative plural)
kantamme (accusative nominative singular)
kantamme (accusative nominative plural)
kantamme (accusative genitive singular)
kantamme (accusative genitive plural)
kantamme (genitive singular)
kantojemme (genitive plural)
kantaimme (genitive plural rare)
kantaamme (partitive singular)
kantojamme (partitive plural)
kannassamme (inessive singular)
kannoissamme (inessive plural)
kannastamme (elative singular)
kannoistamme (elative plural)
kantaamme (illative singular)
kantoihimme (illative plural)
kannallamme (adessive singular)
kannoillamme (adessive plural)
kannaltamme (ablative singular)
kannoiltamme (ablative plural)
kannallemme (allative singular)
kannoillemme (allative plural)
kantanamme (essive singular)
kantoinamme (essive plural)
kannaksemme (singular translative)
kannoiksemme (plural translative)
kannattamme (abessive singular)
kannoittamme (abessive plural)
kantoinemme (comitative plural)
kantanne (nominative singular)
kantanne (nominative plural)
kantanne (accusative nominative singular)
kantanne (accusative nominative plural)
kantanne (accusative genitive singular)
kantanne (accusative genitive plural)
kantanne (genitive singular)
kantojenne (genitive plural)
kantainne (genitive plural rare)
kantaanne (partitive singular)
kantojanne (partitive plural)
kannassanne (inessive singular)
kannoissanne (inessive plural)
kannastanne (elative singular)
kannoistanne (elative plural)
kantaanne (illative singular)
kantoihinne (illative plural)
kannallanne (adessive singular)
kannoillanne (adessive plural)
kannaltanne (ablative singular)
kannoiltanne (ablative plural)
kannallenne (allative singular)
kannoillenne (allative plural)
kantananne (essive singular)
kantoinanne (essive plural)
kannaksenne (singular translative)
kannoiksenne (plural translative)
kannattanne (abessive singular)
kannoittanne (abessive plural)
kantoinenne (comitative plural)
kantansa (nominative singular)
kantansa (nominative plural)
kantansa (accusative nominative singular)
kantansa (accusative nominative plural)
kantansa (accusative genitive singular)
kantansa (accusative genitive plural)
kantansa (genitive singular)
kantojensa (genitive plural)
kantainsa (genitive plural rare)
kantaansa (partitive singular)
kantojaan (partitive plural)
kantojansa (partitive plural)
kannassaan (inessive singular)
kannassansa (inessive singular)
kannoissaan (inessive plural)
kannoissansa (inessive plural)
kannastaan (elative singular)
kannastansa (elative singular)
kannoistaan (elative plural)
kannoistansa (elative plural)
kantaansa (illative singular)
kantoihinsa (illative plural)
kannallaan (adessive singular)
kannallansa (adessive singular)
kannoillaan (adessive plural)
kannoillansa (adessive plural)
kannaltaan (ablative singular)
kannaltansa (ablative singular)
kannoiltaan (ablative plural)
kannoiltansa (ablative plural)
kannalleen (allative singular)
kannallensa (allative singular)
kannoilleen (allative plural)
kannoillensa (allative plural)
kantanaan (essive singular)
kantanansa (essive singular)
kantoinaan (essive plural)
kantoinansa (essive plural)
kannakseen (singular translative)
kannaksensa (singular translative)
kannoikseen (plural translative)
kannoiksensa (plural translative)
kannattaan (abessive singular)
kannattansa (abessive singular)
kannoittaan (abessive plural)
kannoittansa (abessive plural)
kantoineen (comitative plural)
kantoinensa (comitative plural)

[ˈkɑ̝n̪t̪ɑ̝]

From Proto-Finnic *kanta, from Proto-Finno-Ugric *kanta. Cognates include Estonian kand, Erzya кандо (kando, “log, load”), Moksha канда (kanda), Northern Khanty ҳонт (ḥont) and Northern Mansi [script needed] (χānta).


jälki

jälki (nominative singular)
jäljet (nominative plural)
jälki (accusative nominative singular)
jäljet (accusative nominative plural)
jäljen (accusative genitive singular)
jäljet (accusative genitive plural)
jäljen (genitive singular)
jälkien (genitive plural)
jälkeä (partitive singular)
jälkiä (partitive plural)
jäljessä (inessive singular)
jäljissä (inessive plural)
jäljestä (elative singular)
jäljistä (elative plural)
jälkeen (illative singular)
jälkiin (illative plural)
jäljellä (adessive singular)
jäljillä (adessive plural)
jäljeltä (ablative singular)
jäljiltä (ablative plural)
jäljelle (allative singular)
jäljille (allative plural)
jälkenä (essive singular)
jälkinä (essive plural)
jäljeksi (singular translative)
jäljiksi (plural translative)
jäljettä (abessive singular)
jäljittä (abessive plural)
jäljin (instructive plural)
See the possessive forms below. (comitative)
jälkeni (nominative singular)
jälkeni (nominative plural)
jälkeni (accusative nominative singular)
jälkeni (accusative nominative plural)
jälkeni (accusative genitive singular)
jälkeni (accusative genitive plural)
jälkeni (genitive singular)
jälkieni (genitive plural)
jälkeäni (partitive singular)
jälkiäni (partitive plural)
jäljessäni (inessive singular)
jäljissäni (inessive plural)
jäljestäni (elative singular)
jäljistäni (elative plural)
jälkeeni (illative singular)
jälkiini (illative plural)
jäljelläni (adessive singular)
jäljilläni (adessive plural)
jäljeltäni (ablative singular)
jäljiltäni (ablative plural)
jäljelleni (allative singular)
jäljilleni (allative plural)
jälkenäni (essive singular)
jälkinäni (essive plural)
jäljekseni (singular translative)
jäljikseni (plural translative)
jäljettäni (abessive singular)
jäljittäni (abessive plural)
jälkineni (comitative plural)
jälkesi (nominative singular)
jälkesi (nominative plural)
jälkesi (accusative nominative singular)
jälkesi (accusative nominative plural)
jälkesi (accusative genitive singular)
jälkesi (accusative genitive plural)
jälkesi (genitive singular)
jälkiesi (genitive plural)
jälkeäsi (partitive singular)
jälkiäsi (partitive plural)
jäljessäsi (inessive singular)
jäljissäsi (inessive plural)
jäljestäsi (elative singular)
jäljistäsi (elative plural)
jälkeesi (illative singular)
jälkiisi (illative plural)
jäljelläsi (adessive singular)
jäljilläsi (adessive plural)
jäljeltäsi (ablative singular)
jäljiltäsi (ablative plural)
jäljellesi (allative singular)
jäljillesi (allative plural)
jälkenäsi (essive singular)
jälkinäsi (essive plural)
jäljeksesi (singular translative)
jäljiksesi (plural translative)
jäljettäsi (abessive singular)
jäljittäsi (abessive plural)
jälkinesi (comitative plural)
jälkemme (nominative singular)
jälkemme (nominative plural)
jälkemme (accusative nominative singular)
jälkemme (accusative nominative plural)
jälkemme (accusative genitive singular)
jälkemme (accusative genitive plural)
jälkemme (genitive singular)
jälkiemme (genitive plural)
jälkeämme (partitive singular)
jälkiämme (partitive plural)
jäljessämme (inessive singular)
jäljissämme (inessive plural)
jäljestämme (elative singular)
jäljistämme (elative plural)
jälkeemme (illative singular)
jälkiimme (illative plural)
jäljellämme (adessive singular)
jäljillämme (adessive plural)
jäljeltämme (ablative singular)
jäljiltämme (ablative plural)
jäljellemme (allative singular)
jäljillemme (allative plural)
jälkenämme (essive singular)
jälkinämme (essive plural)
jäljeksemme (singular translative)
jäljiksemme (plural translative)
jäljettämme (abessive singular)
jäljittämme (abessive plural)
jälkinemme (comitative plural)
jälkenne (nominative singular)
jälkenne (nominative plural)
jälkenne (accusative nominative singular)
jälkenne (accusative nominative plural)
jälkenne (accusative genitive singular)
jälkenne (accusative genitive plural)
jälkenne (genitive singular)
jälkienne (genitive plural)
jälkeänne (partitive singular)
jälkiänne (partitive plural)
jäljessänne (inessive singular)
jäljissänne (inessive plural)
jäljestänne (elative singular)
jäljistänne (elative plural)
jälkeenne (illative singular)
jälkiinne (illative plural)
jäljellänne (adessive singular)
jäljillänne (adessive plural)
jäljeltänne (ablative singular)
jäljiltänne (ablative plural)
jäljellenne (allative singular)
jäljillenne (allative plural)
jälkenänne (essive singular)
jälkinänne (essive plural)
jäljeksenne (singular translative)
jäljiksenne (plural translative)
jäljettänne (abessive singular)
jäljittänne (abessive plural)
jälkinenne (comitative plural)
jälkensä (nominative singular)
jälkensä (nominative plural)
jälkensä (accusative nominative singular)
jälkensä (accusative nominative plural)
jälkensä (accusative genitive singular)
jälkensä (accusative genitive plural)
jälkensä (genitive singular)
jälkiensä (genitive plural)
jälkeään (partitive singular)
jälkeänsä (partitive singular)
jälkiään (partitive plural)
jälkiänsä (partitive plural)
jäljessään (inessive singular)
jäljessänsä (inessive singular)
jäljissään (inessive plural)
jäljissänsä (inessive plural)
jäljestään (elative singular)
jäljestänsä (elative singular)
jäljistään (elative plural)
jäljistänsä (elative plural)
jälkeensä (illative singular)
jälkiinsä (illative plural)
jäljellään (adessive singular)
jäljellänsä (adessive singular)
jäljillään (adessive plural)
jäljillänsä (adessive plural)
jäljeltään (ablative singular)
jäljeltänsä (ablative singular)
jäljiltään (ablative plural)
jäljiltänsä (ablative plural)
jäljelleen (allative singular)
jäljellensä (allative singular)
jäljilleen (allative plural)
jäljillensä (allative plural)
jälkenään (essive singular)
jälkenänsä (essive singular)
jälkinään (essive plural)
jälkinänsä (essive plural)
jäljekseen (singular translative)
jäljeksensä (singular translative)
jäljikseen (plural translative)
jäljiksensä (plural translative)
jäljettään (abessive singular)
jäljettänsä (abessive singular)
jäljittään (abessive plural)
jäljittänsä (abessive plural)
jälkineen (comitative plural)
jälkinensä (comitative plural)

From Proto-Finnic *jälki, from Proto-Uralic *jelkä (“trace”). Cognate with Estonian jälg, Hungarian jel and possibly Erzya эльге (eľge) as well.

(linear algebra) trace (of a square matrix)


aste

aste (nominative singular)
asteet (nominative plural)
aste (accusative nominative singular)
asteet (accusative nominative plural)
asteen (accusative genitive singular)
asteet (accusative genitive plural)
asteen (genitive singular)
asteiden (genitive plural)
asteitten (genitive plural)
astetta (partitive singular)
asteita (partitive plural)
asteessa (inessive singular)
asteissa (inessive plural)
asteesta (elative singular)
asteista (elative plural)
asteeseen (illative singular)
asteisiin (illative plural)
asteihin (illative plural)
asteella (adessive singular)
asteilla (adessive plural)
asteelta (ablative singular)
asteilta (ablative plural)
asteelle (allative singular)
asteille (allative plural)
asteena (essive singular)
asteina (essive plural)
asteeksi (singular translative)
asteiksi (plural translative)
asteetta (abessive singular)
asteitta (abessive plural)
astein (instructive plural)
See the possessive forms below. (comitative)
asteeni (nominative singular)
asteeni (nominative plural)
asteeni (accusative nominative singular)
asteeni (accusative nominative plural)
asteeni (accusative genitive singular)
asteeni (accusative genitive plural)
asteeni (genitive singular)
asteideni (genitive plural)
asteitteni (genitive plural)
astettani (partitive singular)
asteitani (partitive plural)
asteessani (inessive singular)
asteissani (inessive plural)
asteestani (elative singular)
asteistani (elative plural)
asteeseeni (illative singular)
asteisiini (illative plural)
asteihini (illative plural)
asteellani (adessive singular)
asteillani (adessive plural)
asteeltani (ablative singular)
asteiltani (ablative plural)
asteelleni (allative singular)
asteilleni (allative plural)
asteenani (essive singular)
asteinani (essive plural)
asteekseni (singular translative)
asteikseni (plural translative)
asteettani (abessive singular)
asteittani (abessive plural)
asteineni (comitative plural)
asteesi (nominative singular)
asteesi (nominative plural)
asteesi (accusative nominative singular)
asteesi (accusative nominative plural)
asteesi (accusative genitive singular)
asteesi (accusative genitive plural)
asteesi (genitive singular)
asteidesi (genitive plural)
asteittesi (genitive plural)
astettasi (partitive singular)
asteitasi (partitive plural)
asteessasi (inessive singular)
asteissasi (inessive plural)
asteestasi (elative singular)
asteistasi (elative plural)
asteeseesi (illative singular)
asteisiisi (illative plural)
asteihisi (illative plural)
asteellasi (adessive singular)
asteillasi (adessive plural)
asteeltasi (ablative singular)
asteiltasi (ablative plural)
asteellesi (allative singular)
asteillesi (allative plural)
asteenasi (essive singular)
asteinasi (essive plural)
asteeksesi (singular translative)
asteiksesi (plural translative)
asteettasi (abessive singular)
asteittasi (abessive plural)
asteinesi (comitative plural)
asteemme (nominative singular)
asteemme (nominative plural)
asteemme (accusative nominative singular)
asteemme (accusative nominative plural)
asteemme (accusative genitive singular)
asteemme (accusative genitive plural)
asteemme (genitive singular)
asteidemme (genitive plural)
asteittemme (genitive plural)
astettamme (partitive singular)
asteitamme (partitive plural)
asteessamme (inessive singular)
asteissamme (inessive plural)
asteestamme (elative singular)
asteistamme (elative plural)
asteeseemme (illative singular)
asteisiimme (illative plural)
asteihimme (illative plural)
asteellamme (adessive singular)
asteillamme (adessive plural)
asteeltamme (ablative singular)
asteiltamme (ablative plural)
asteellemme (allative singular)
asteillemme (allative plural)
asteenamme (essive singular)
asteinamme (essive plural)
asteeksemme (singular translative)
asteiksemme (plural translative)
asteettamme (abessive singular)
asteittamme (abessive plural)
asteinemme (comitative plural)
asteenne (nominative singular)
asteenne (nominative plural)
asteenne (accusative nominative singular)
asteenne (accusative nominative plural)
asteenne (accusative genitive singular)
asteenne (accusative genitive plural)
asteenne (genitive singular)
asteidenne (genitive plural)
asteittenne (genitive plural)
astettanne (partitive singular)
asteitanne (partitive plural)
asteessanne (inessive singular)
asteissanne (inessive plural)
asteestanne (elative singular)
asteistanne (elative plural)
asteeseenne (illative singular)
asteisiinne (illative plural)
asteihinne (illative plural)
asteellanne (adessive singular)
asteillanne (adessive plural)
asteeltanne (ablative singular)
asteiltanne (ablative plural)
asteellenne (allative singular)
asteillenne (allative plural)
asteenanne (essive singular)
asteinanne (essive plural)
asteeksenne (singular translative)
asteiksenne (plural translative)
asteettanne (abessive singular)
asteittanne (abessive plural)
asteinenne (comitative plural)
asteensa (nominative singular)
asteensa (nominative plural)
asteensa (accusative nominative singular)
asteensa (accusative nominative plural)
asteensa (accusative genitive singular)
asteensa (accusative genitive plural)
asteensa (genitive singular)
asteidensa (genitive plural)
asteittensa (genitive plural)
astettaan (partitive singular)
astettansa (partitive singular)
asteitaan (partitive plural)
asteitansa (partitive plural)
asteessaan (inessive singular)
asteessansa (inessive singular)
asteissaan (inessive plural)
asteissansa (inessive plural)
asteestaan (elative singular)
asteestansa (elative singular)
asteistaan (elative plural)
asteistansa (elative plural)
asteeseensa (illative singular)
asteisiinsa (illative plural)
asteihinsa (illative plural)
asteellaan (adessive singular)
asteellansa (adessive singular)
asteillaan (adessive plural)
asteillansa (adessive plural)
asteeltaan (ablative singular)
asteeltansa (ablative singular)
asteiltaan (ablative plural)
asteiltansa (ablative plural)
asteelleen (allative singular)
asteellensa (allative singular)
asteilleen (allative plural)
asteillensa (allative plural)
asteenaan (essive singular)
asteenansa (essive singular)
asteinaan (essive plural)
asteinansa (essive plural)
asteekseen (singular translative)
asteeksensa (singular translative)
asteikseen (plural translative)
asteiksensa (plural translative)
asteettaan (abessive singular)
asteettansa (abessive singular)
asteittaan (abessive plural)
asteittansa (abessive plural)
asteineen (comitative plural)
asteinensa (comitative plural)

→ Livvi: aste

astua (“to step”) + -e. Coined by Finnish physician and translator Wolmar Schildt in 1846, based on dialectal "step (in stairs)", cases of which was also used as adverbs with more figurative meanings like those currently used.

coin

(linear algebra) rank (maximal number of linearly independent columns or rows of a matrix)


ket

From bra-ket notation invented by Paul Dirac, from bracket.

A particular ket, say #124;A#92;rangle, might be represented by a particular column vector. Its corresponding bra, #92;langleA#124;, would then be represented by the row vector which is the transpose conjugate of that column vector.

(linear algebra, quantum mechanics) A column vector, in Hilbert space, especially as representing the state of a quantum mechanical system; the complex conjugate transpose of a bra (a row vector); a ket vector. Symbolised by |...〉.


adjoint

/ˈæd͡ʒ.ɔɪnt/

From French adjoindre (“to join”), from late 19th C; see also adjoin. Doublet of adjunct.
In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with adjunct and the relationship is called an adjunction. The origin of any particular usage may therefore be uncertain.

(mathematics, linear algebra, of a matrix) Transpose conjugate.


determinan

détèrminan (canonical)
determinan-determinan (plural)

Internationalism, borrowed from Dutch determinant.

(Standard Malay)

(linear algebra) a scalar that encodes certain characteristics of a given transformation matrix; the unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value 1 for the unit matrix


invertible matrix

en-noun

invertible matrix (plural invertible matrices)

1975 [Prentice-Hall], Kenneth Hoffman, Analysis in Euclidean Space, Dover, 2007, page 65,
It says that, if A is a singular matrix, then every neighborhood of A contains an invertible matrix. In other words, if A is singular, we can perturb A just a little and obtain an invertible matrix.

There are certain very simple invertible matrices, and every invertible matrix over a field can be built up out of them.

1997, Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, page 199:

The space of real square matrices of fixed size is a vector space whose dimension is a perfect square and the invertible matrices constitute a dense open subset of this vector space.

2013, Mahya Ghandehari, Aizhan Syzdykova, Keith F. Taylor, “A four dimensional continuous wavelet transform”, in Azita Mayeli, editor, Commutative and Noncommutative Harmonic Analysis and Applications, American Mathematical Society, page 123:

(linear algebra) Any n×n square matrix for which there exists a corresponding inverse matrix (i.e., a second (or possibly the same) matrix such that when the two are multiplied by each other, in either order, the result is the n×n identity matrix).

kääntyvä matriisi

square matrix which, when multiplied by some other, yields the identity matrix


matriisi

matriisi (nominative singular)
matriisit (nominative plural)
matriisi (accusative nominative singular)
matriisit (accusative nominative plural)
matriisin (accusative genitive singular)
matriisit (accusative genitive plural)
matriisin (genitive singular)
matriisien (genitive plural)
matriisia (partitive singular)
matriiseja (partitive plural)
matriisissa (inessive singular)
matriiseissa (inessive plural)
matriisista (elative singular)
matriiseista (elative plural)
matriisiin (illative singular)
matriiseihin (illative plural)
matriisilla (adessive singular)
matriiseilla (adessive plural)
matriisilta (ablative singular)
matriiseilta (ablative plural)
matriisille (allative singular)
matriiseille (allative plural)
matriisina (essive singular)
matriiseina (essive plural)
matriisiksi (singular translative)
matriiseiksi (plural translative)
matriisitta (abessive singular)
matriiseitta (abessive plural)
matriisein (instructive plural)
See the possessive forms below. (comitative)
matriisini (nominative singular)
matriisini (nominative plural)
matriisini (accusative nominative singular)
matriisini (accusative nominative plural)
matriisini (accusative genitive singular)
matriisini (accusative genitive plural)
matriisini (genitive singular)
matriisieni (genitive plural)
matriisiani (partitive singular)
matriisejani (partitive plural)
matriisissani (inessive singular)
matriiseissani (inessive plural)
matriisistani (elative singular)
matriiseistani (elative plural)
matriisiini (illative singular)
matriiseihini (illative plural)
matriisillani (adessive singular)
matriiseillani (adessive plural)
matriisiltani (ablative singular)
matriiseiltani (ablative plural)
matriisilleni (allative singular)
matriiseilleni (allative plural)
matriisinani (essive singular)
matriiseinani (essive plural)
matriisikseni (singular translative)
matriiseikseni (plural translative)
matriisittani (abessive singular)
matriiseittani (abessive plural)
matriiseineni (comitative plural)
matriisisi (nominative singular)
matriisisi (nominative plural)
matriisisi (accusative nominative singular)
matriisisi (accusative nominative plural)
matriisisi (accusative genitive singular)
matriisisi (accusative genitive plural)
matriisisi (genitive singular)
matriisiesi (genitive plural)
matriisiasi (partitive singular)
matriisejasi (partitive plural)
matriisissasi (inessive singular)
matriiseissasi (inessive plural)
matriisistasi (elative singular)
matriiseistasi (elative plural)
matriisiisi (illative singular)
matriiseihisi (illative plural)
matriisillasi (adessive singular)
matriiseillasi (adessive plural)
matriisiltasi (ablative singular)
matriiseiltasi (ablative plural)
matriisillesi (allative singular)
matriiseillesi (allative plural)
matriisinasi (essive singular)
matriiseinasi (essive plural)
matriisiksesi (singular translative)
matriiseiksesi (plural translative)
matriisittasi (abessive singular)
matriiseittasi (abessive plural)
matriiseinesi (comitative plural)
matriisimme (nominative singular)
matriisimme (nominative plural)
matriisimme (accusative nominative singular)
matriisimme (accusative nominative plural)
matriisimme (accusative genitive singular)
matriisimme (accusative genitive plural)
matriisimme (genitive singular)
matriisiemme (genitive plural)
matriisiamme (partitive singular)
matriisejamme (partitive plural)
matriisissamme (inessive singular)
matriiseissamme (inessive plural)
matriisistamme (elative singular)
matriiseistamme (elative plural)
matriisiimme (illative singular)
matriiseihimme (illative plural)
matriisillamme (adessive singular)
matriiseillamme (adessive plural)
matriisiltamme (ablative singular)
matriiseiltamme (ablative plural)
matriisillemme (allative singular)
matriiseillemme (allative plural)
matriisinamme (essive singular)
matriiseinamme (essive plural)
matriisiksemme (singular translative)
matriiseiksemme (plural translative)
matriisittamme (abessive singular)
matriiseittamme (abessive plural)
matriiseinemme (comitative plural)
matriisinne (nominative singular)
matriisinne (nominative plural)
matriisinne (accusative nominative singular)
matriisinne (accusative nominative plural)
matriisinne (accusative genitive singular)
matriisinne (accusative genitive plural)
matriisinne (genitive singular)
matriisienne (genitive plural)
matriisianne (partitive singular)
matriisejanne (partitive plural)
matriisissanne (inessive singular)
matriiseissanne (inessive plural)
matriisistanne (elative singular)
matriiseistanne (elative plural)
matriisiinne (illative singular)
matriiseihinne (illative plural)
matriisillanne (adessive singular)
matriiseillanne (adessive plural)
matriisiltanne (ablative singular)
matriiseiltanne (ablative plural)
matriisillenne (allative singular)
matriiseillenne (allative plural)
matriisinanne (essive singular)
matriiseinanne (essive plural)
matriisiksenne (singular translative)
matriiseiksenne (plural translative)
matriisittanne (abessive singular)
matriiseittanne (abessive plural)
matriiseinenne (comitative plural)
matriisinsa (nominative singular)
matriisinsa (nominative plural)
matriisinsa (accusative nominative singular)
matriisinsa (accusative nominative plural)
matriisinsa (accusative genitive singular)
matriisinsa (accusative genitive plural)
matriisinsa (genitive singular)
matriisiensa (genitive plural)
matriisiaan (partitive singular)
matriisiansa (partitive singular)
matriisejaan (partitive plural)
matriisejansa (partitive plural)
matriisissaan (inessive singular)
matriisissansa (inessive singular)
matriiseissaan (inessive plural)
matriiseissansa (inessive plural)
matriisistaan (elative singular)
matriisistansa (elative singular)
matriiseistaan (elative plural)
matriiseistansa (elative plural)
matriisiinsa (illative singular)
matriiseihinsa (illative plural)
matriisillaan (adessive singular)
matriisillansa (adessive singular)
matriiseillaan (adessive plural)
matriiseillansa (adessive plural)
matriisiltaan (ablative singular)
matriisiltansa (ablative singular)
matriiseiltaan (ablative plural)
matriiseiltansa (ablative plural)
matriisilleen (allative singular)
matriisillensa (allative singular)
matriiseilleen (allative plural)
matriiseillensa (allative plural)
matriisinaan (essive singular)
matriisinansa (essive singular)
matriiseinaan (essive plural)
matriiseinansa (essive plural)
matriisikseen (singular translative)
matriisiksensa (singular translative)
matriiseikseen (plural translative)
matriiseiksensa (plural translative)
matriisittaan (abessive singular)
matriisittansa (abessive singular)
matriiseittaan (abessive plural)
matriiseittansa (abessive plural)
matriiseineen (comitative plural)
matriiseinensa (comitative plural)

Borrowed from Swedish matris, from French matrice.

(technology, mathematics, computing) matrix


virittää

viritän (first-person indicative present singular)
en viritä (first-person indicative negative present singular)
olen virittänyt (first-person indicative perfect singular)
en ole virittänyt (first-person indicative negative perfect singular)
virität (indicative present second-person singular)
et viritä (indicative negative present second-person singular)
olet virittänyt (indicative perfect second-person singular)
et ole virittänyt (indicative negative perfect second-person singular)
virittää (indicative present singular third-person)
ei viritä (indicative negative present singular third-person)
on virittänyt (indicative perfect singular third-person)
ei ole virittänyt (indicative negative perfect singular third-person)
viritämme (first-person indicative plural present)
emme viritä (first-person indicative negative plural present)
olemme virittäneet (first-person indicative perfect plural)
emme ole virittäneet (first-person indicative negative perfect plural)
viritätte (indicative plural present second-person)
ette viritä (indicative negative plural present second-person)
olette virittäneet (indicative perfect plural second-person)
ette ole virittäneet (indicative negative perfect plural second-person)
virittävät (indicative plural present third-person)
eivät viritä (indicative negative plural present third-person)
ovat virittäneet (indicative perfect plural third-person)
eivät ole virittäneet (indicative negative perfect plural third-person)
viritetään (indicative passive present)
ei viritetä (indicative negative passive present)
on viritetty (indicative passive perfect)
ei ole viritetty (indicative negative passive perfect)
viritin (first-person indicative past singular)
en virittänyt (first-person indicative negative past singular)
olin virittänyt (first-person indicative pluperfect singular)
en ollut virittänyt (first-person indicative negative pluperfect singular)
viritit (indicative past second-person singular)
et virittänyt (indicative negative past second-person singular)
olit virittänyt (indicative pluperfect second-person singular)
et ollut virittänyt (indicative negative pluperfect second-person singular)
viritti (indicative past singular third-person)
ei virittänyt (indicative negative past singular third-person)
oli virittänyt (indicative pluperfect singular third-person)
ei ollut virittänyt (indicative negative pluperfect singular third-person)
viritimme (first-person indicative past plural)
emme virittäneet (first-person indicative negative past plural)
olimme virittäneet (first-person indicative pluperfect plural)
emme olleet virittäneet (first-person indicative negative pluperfect plural)
virititte (indicative past plural second-person)
ette virittäneet (indicative negative past plural second-person)
olitte virittäneet (indicative pluperfect plural second-person)
ette olleet virittäneet (indicative negative pluperfect plural second-person)
virittivät (indicative past plural third-person)
eivät virittäneet (indicative negative past plural third-person)
olivat virittäneet (indicative pluperfect plural third-person)
eivät olleet virittäneet (indicative negative pluperfect plural third-person)
viritettiin (indicative passive past)
ei viritetty (indicative negative passive past)
oli viritetty (indicative passive pluperfect)
ei ollut viritetty (indicative negative passive pluperfect)
virittäisin (conditional first-person present singular)
en virittäisi (conditional first-person negative present singular)
olisin virittänyt (conditional first-person perfect singular)
en olisi virittänyt (conditional first-person negative perfect singular)
virittäisit (conditional present second-person singular)
et virittäisi (conditional negative present second-person singular)
olisit virittänyt (conditional perfect second-person singular)
et olisi virittänyt (conditional negative perfect second-person singular)
virittäisi (conditional present singular third-person)
ei virittäisi (conditional negative present singular third-person)
olisi virittänyt (conditional perfect singular third-person)
ei olisi virittänyt (conditional negative perfect singular third-person)
virittäisimme (conditional first-person plural present)
emme virittäisi (conditional first-person negative plural present)
olisimme virittäneet (conditional first-person perfect plural)
emme olisi virittäneet (conditional first-person negative perfect plural)
virittäisitte (conditional plural present second-person)
ette virittäisi (conditional negative plural present second-person)
olisitte virittäneet (conditional perfect plural second-person)
ette olisi virittäneet (conditional negative perfect plural second-person)
virittäisivät (conditional plural present third-person)
eivät virittäisi (conditional negative plural present third-person)
olisivat virittäneet (conditional perfect plural third-person)
eivät olisi virittäneet (conditional negative perfect plural third-person)
viritettäisiin (conditional passive present)
ei viritettäisi (conditional negative passive present)
olisi viritetty (conditional passive perfect)
ei olisi viritetty (conditional negative passive perfect)
viritä (imperative present second-person singular)
älä viritä (imperative negative present second-person singular)
virittäköön (imperative present singular third-person)
älköön virittäkö (imperative negative present singular third-person)
olkoon virittänyt (imperative perfect singular third-person)
älköön olko virittänyt (imperative negative perfect singular third-person)
virittäkäämme (first-person imperative plural present)
älkäämme virittäkö (first-person imperative negative plural present)
virittäkää (imperative plural present second-person)
älkää virittäkö (imperative negative plural present second-person)
virittäkööt (imperative plural present third-person)
älkööt virittäkö (imperative negative plural present third-person)
olkoot virittäneet (imperative perfect plural third-person)
älkööt olko virittäneet (imperative negative perfect plural third-person)
viritettäköön (imperative passive present)
älköön viritettäkö (imperative negative passive present)
olkoon viritetty (imperative passive perfect)
älköön olko viritetty (imperative negative passive perfect)
virittänen (first-person potential present singular)
en virittäne (first-person negative potential present singular)
lienen virittänyt (first-person perfect potential singular)
en liene virittänyt (first-person negative perfect potential singular)
virittänet (potential present second-person singular)
et virittäne (negative potential present second-person singular)
lienet virittänyt (perfect potential second-person singular)
et liene virittänyt (negative perfect potential second-person singular)
virittänee (potential present singular third-person)
ei virittäne (negative potential present singular third-person)
lienee virittänyt (perfect potential singular third-person)
ei liene virittänyt (negative perfect potential singular third-person)
virittänemme (first-person plural potential present)
emme virittäne (first-person negative plural potential present)
lienemme virittäneet (first-person perfect plural potential)
emme liene virittäneet (first-person negative perfect plural potential)
virittänette (plural potential present second-person)
ette virittäne (negative plural potential present second-person)
lienette virittäneet (perfect plural potential second-person)
ette liene virittäneet (negative perfect plural potential second-person)
virittänevät (plural potential present third-person)
eivät virittäne (negative plural potential present third-person)
lienevät virittäneet (perfect plural potential third-person)
eivät liene virittäneet (negative perfect plural potential third-person)
viritettäneen (passive potential present)
ei viritettäne (negative passive potential present)
lienee viritetty (passive perfect potential)
ei liene viritetty (negative passive perfect potential)
virittää (infinitive infinitive-i)
virittävä (active participle present)
viritettävä (participle passive present)
virittääkseni (first-person possessive singular)
virittääksemme (first-person plural possessive)
virittääksesi (possessive second-person singular)
virittääksenne (plural possessive second-person)
virittääkseen (possessive third-person)
virittääksensä (possessive third-person)
virittänyt (infinitive-i-long past)
viritetty (infinitive-i-long past)
virittäessä (inessive second-person)
viritettäessä (inessive second-person)
virittämä (agent)
virittäessäni (first-person possessive singular)
virittäessämme (first-person plural possessive)
virittäessäsi (possessive second-person singular)
virittäessänne (plural possessive second-person)
virittäessään (possessive third-person)
virittäessänsä (possessive third-person)
virittämätön (inessive negative second-person)
virittäen (instructive second-person)
virittämässä (inessive third-person)
virittämästä (elative third-person)
virittämään (illative third-person)
virittämällä (adessive third-person)
virittämättä (abessive third-person)
virittämän (instructive third-person)
viritettämän (instructive third-person)
virittäminen (fourth-person noun-from-verb)
virittämäisilläni (first-person possessive singular)
virittämäisillämme (first-person plural possessive)
virittämäisilläsi (possessive second-person singular)
virittämäisillänne (plural possessive second-person)
virittämäisillään (possessive third-person)
virittämäisillänsä (possessive third-person)

/ˈʋiritːæːˣ/

From Proto-Finnic *virittädäk, equivalent to vireä + -ittää. Related to Karelian viritteä, Votic virittää (Kattila) and possibly Erzya верьгедемс (veŕgedems, “to ignite”).

cog

(linear algebra) to span (generate an entire space by means of linear combinations)


linear combination

en-noun

linear combination (plural linear combinations)

(linear algebra) a sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element)


linearly independent

(linear algebra) (Of a set of vectors or ring elements) whose nontrivial linear combinations are nonzero


characteristic polynomial

characteristic polynomial (plural characteristic polynomials)

The characteristic polynomial of (cc 1 4\3-5) is |cc 1-x 4\3-5-x|=x²+4x-17.

The characteristic polynomial of a 2×2 matrix M is λ²- mbox tr(M)λ+ mbox det(M), where mbox tr(M) denotes the trace of M and mbox det(M) denotes the determinant of M.

The characteristic polynomial of a 3×3 matrix M is -λ³+ mbox tr(M)λ²- mbox tr( mbox adj(M))λ+ mbox det(M), where mbox adj(M) denotes the adjugate of M.

(linear algebra) The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant.


linear transformation

linear transformation (plural linear transformations)

(linear algebra) A map between vector spaces which preserves the operations of vector addition and scalar multiplication.

lineêre transformasie

a map between vector spaces preserving vector addition and scalar multiplication


skew-symmetric

(ccc 0 2 3\-2 0-4\-3 4 0)

(linear algebra) Of a matrix, satisfying A^( textsf )T=-A, i.e. having entries on one side of the diagonal that are the additive inverses of their correspondents on the other side of the diagonal and having only zeroes on the main diagonal.

schiefsymmetrisch

whose entries on one side of the diagonal are the additive inverses of their correspondents on the other side of the diagonal and whose elements on the main diagonal are zero


skew-symmetry

(linear algebra) The state of being skew-symmetric


unit vector

(linear algebra, mathematical analysis) A vector with the length of 1, defined as mathbf ̂u=(mathbf u)/(‖ mathbf u‖).


overdetermined

en-adj

overdetermined (comparative more overdetermined, superlative most overdetermined)

more overdetermined (comparative)
most overdetermined (superlative)

LL-Q1860 (eng)-Flame, not lame-overdetermined.wav

(linear algebra, of a system of linear equations) Having more equations than variables.


正定値行列

ja-noun

正(せい)定(てい)値(ち)行(ぎょう)列(れつ) • (seiteichi gyōretsu)

正定値行列 (canonical)
seiteichi gyōretsu (romanization)

positive-definite matrix
positive-definite matrix

(linear algebra) positive-definite matrix


半正定値行列

はんせいていち ぎょうれつ

半(はん)正(せい)定(てい)値(ち)行(ぎょう)列(れつ) • (hanseiteichi gyōretsu)

半正定値行列 (canonical)
hanseiteichi gyōretsu (romanization)

positive-semidefinite matrix
positive-semidefinite matrix

(linear algebra) positive-semidefinite matrix


エルミート行列

^エルミート ぎょうれつ

エルミート行列 (canonical)
Erumīto gyōretsu (romanization)


転置

ja-verb-suru

転(てん)置(ち)する • (tenchi suru) suru (stem 転(てん)置(ち)し (tenchi shi), past 転(てん)置(ち)した (tenchi shita))

転置する (canonical)
tenchi suru (romanization)
転置し (stem)
転置した (past)


固有値

(linear algebra) eigenvalue, characteristic value


転置行列

ja-noun

転(てん)置(ち)行(ぎょう)列(れつ) • (tenchi gyōretsu)

転置行列 (canonical)
tenchi gyōretsu (romanization)


正方行列

正(せい)方(ほう)行(ぎょう)列(れつ) • (seihō gyōretsu) ^(←せいはうぎやうれつ (seifau gyauretu)?)

正方行列 (canonical)
seihō gyōretsu (romanization)
seifau gyauretu (romanization)
せいはうぎやうれつ (hiragana historical)

[se̞ːho̞ː ɡʲo̞ːɾe̞t͡sɨ]


三角行列

三(さん)角(かく)行(ぎょう)列(れつ) • (sankaku gyōretsu)

三角行列 (canonical)
sankaku gyōretsu (romanization)


上三角行列

うえ さんかく ぎょうれつ

上(うえ)三(さん)角(かく)行(ぎょう)列(れつ) • (ue sankaku gyōretsu)

上三角行列 (canonical)
ue sankaku gyōretsu (romanization)

upper triangular matrix
upper triangular matrix

(linear algebra) upper triangular matrix


下三角行列

した さんかく ぎょうれつ

下(した)三(さん)角(かく)行(ぎょう)列(れつ) • (shita sankaku gyōretsu)

下三角行列 (canonical)
shita sankaku gyōretsu (romanization)

lower triangular matrix
lower triangular matrix

(linear algebra) lower triangular matrix


対角行列

対(たい)角(かく)行(ぎょう)列(れつ) • (taikaku gyōretsu)

対角行列 (canonical)
taikaku gyōretsu (romanization)

対角成分

対角成分 たいかくせいぶん


対角成分

対角成分 (canonical)
taikaku seibun (romanization)

(linear algebra) diagonal element; diagonal component


対称行列

対(たい)称(しょう)行(ぎょう)列(れつ) • (taishō gyōretsu)

対称行列 (canonical)
taishō gyōretsu (romanization)

(linear algebra, mathematics) symmetric matrix


交代行列

交(こう)代(たい)行(ぎょう)列(れつ) • (kōtai gyōretsu)

交代行列 (canonical)
kōtai gyōretsu (romanization)

skew-symmetric matrix
skew-symmetric matrix


逆行列

逆行列 (canonical)
gyaku gyōretsu (romanization)


外積

/waɪ̯⁵¹ t͡ɕi⁵⁵/

/ŋɔːi̯²² t͡sɪk̚⁵/

/waɪ̯⁵¹ t͡ɕi⁵⁵/

/ŋɔːi̯²² t͡sɪk̚⁵/

(linear algebra) outer product; cross product


単位行列

単位行列 (canonical)
tan'i gyōretsu (romanization)


行ベクトル

行ベクトル (canonical)
gyō bekutoru (romanization)


列ベクトル

列ベクトル (canonical)
retsu bekutoru (romanization)


contravariant

LL-Q1860 (eng)-Flame, not lame-contravariant.wav

(linear algebra, of a vector) Scaling inversely with a change of basis.


row-equivalence

equivalence relation
equivalence relation

(linear algebra) A relation between two matrices of the same size, such that every row of one matrix is a linear combination of the rows of the other matrix, and vice versa. It is an equivalence relation.


eigingildi

is-noun

eigingildi n (genitive singular eigingildis, nominative plural eigingildi)

eigingildis (genitive singular)
eigingildi (nominative plural)
eigingildi (indefinite nominative singular)
eigingildið (definite nominative singular)
eigingildi (indefinite nominative plural)
eigingildin (definite nominative plural)
eigingildi (accusative indefinite singular)
eigingildið (accusative definite singular)
eigingildi (accusative indefinite plural)
eigingildin (accusative definite plural)
eigingildi (dative indefinite singular)
eigingildinu (dative definite singular)
eigingildum (dative indefinite plural)
eigingildunum (dative definite plural)
eigingildis (genitive indefinite singular)
eigingildisins (definite genitive singular)
eigingilda (genitive indefinite plural)
eigingildanna (definite genitive plural)

From eigin- (“eigen-”) + gildi (“value”).

characteristic value
characteristic value

(mathematics, linear algebra) eigenvalue, characteristic value


lineaarialgebra

lineaarialgebra (nominative singular)
lineaarialgebrat (nominative plural)
lineaarialgebra (accusative nominative singular)
lineaarialgebrat (accusative nominative plural)
lineaarialgebran (accusative genitive singular)
lineaarialgebrat (accusative genitive plural)
lineaarialgebran (genitive singular)
lineaarialgebrien (genitive plural)
lineaarialgebroiden (genitive plural)
lineaarialgebroitten (genitive plural)
lineaarialgebrojen (genitive plural rare)
lineaarialgebrain (genitive plural rare)
lineaarialgebraa (partitive singular)
lineaarialgebria (partitive plural)
lineaarialgebroita (partitive plural)
lineaarialgebroja (partitive plural rare)
lineaarialgebrassa (inessive singular)
lineaarialgebroissa (inessive plural)
lineaarialgebrissa (inessive plural)
lineaarialgebrasta (elative singular)
lineaarialgebroista (elative plural)
lineaarialgebrista (elative plural)
lineaarialgebraan (illative singular)
lineaarialgebriin (illative plural)
lineaarialgebroihin (illative plural)
lineaarialgebralla (adessive singular)
lineaarialgebroilla (adessive plural)
lineaarialgebrilla (adessive plural)
lineaarialgebralta (ablative singular)
lineaarialgebroilta (ablative plural)
lineaarialgebrilta (ablative plural)
lineaarialgebralle (allative singular)
lineaarialgebroille (allative plural)
lineaarialgebrille (allative plural)
lineaarialgebrana (essive singular)
lineaarialgebroina (essive plural)
lineaarialgebrina (essive plural)
lineaarialgebraksi (singular translative)
lineaarialgebroiksi (plural translative)
lineaarialgebriksi (plural translative)
lineaarialgebratta (abessive singular)
lineaarialgebroitta (abessive plural)
lineaarialgebritta (abessive plural)
lineaarialgebroin (instructive plural)
lineaarialgebrin (instructive plural)
See the possessive forms below. (comitative)
lineaarialgebrani (nominative singular)
lineaarialgebrani (nominative plural)
lineaarialgebrani (accusative nominative singular)
lineaarialgebrani (accusative nominative plural)
lineaarialgebrani (accusative genitive singular)
lineaarialgebrani (accusative genitive plural)
lineaarialgebrani (genitive singular)
lineaarialgebrieni (genitive plural)
lineaarialgebroideni (genitive plural)
lineaarialgebroitteni (genitive plural)
lineaarialgebrojeni (genitive plural rare)
lineaarialgebraini (genitive plural rare)
lineaarialgebraani (partitive singular)
lineaarialgebriani (partitive plural)
lineaarialgebroitani (partitive plural)
lineaarialgebrojani (partitive plural rare)
lineaarialgebrassani (inessive singular)
lineaarialgebroissani (inessive plural)
lineaarialgebrissani (inessive plural)
lineaarialgebrastani (elative singular)
lineaarialgebroistani (elative plural)
lineaarialgebristani (elative plural)
lineaarialgebraani (illative singular)
lineaarialgebriini (illative plural)
lineaarialgebroihini (illative plural)
lineaarialgebrallani (adessive singular)
lineaarialgebroillani (adessive plural)
lineaarialgebrillani (adessive plural)
lineaarialgebraltani (ablative singular)
lineaarialgebroiltani (ablative plural)
lineaarialgebriltani (ablative plural)
lineaarialgebralleni (allative singular)
lineaarialgebroilleni (allative plural)
lineaarialgebrilleni (allative plural)
lineaarialgebranani (essive singular)
lineaarialgebroinani (essive plural)
lineaarialgebrinani (essive plural)
lineaarialgebrakseni (singular translative)
lineaarialgebroikseni (plural translative)
lineaarialgebrikseni (plural translative)
lineaarialgebrattani (abessive singular)
lineaarialgebroittani (abessive plural)
lineaarialgebrittani (abessive plural)
lineaarialgebroineni (comitative plural)
lineaarialgebrineni (comitative plural)
lineaarialgebrasi (nominative singular)
lineaarialgebrasi (nominative plural)
lineaarialgebrasi (accusative nominative singular)
lineaarialgebrasi (accusative nominative plural)
lineaarialgebrasi (accusative genitive singular)
lineaarialgebrasi (accusative genitive plural)
lineaarialgebrasi (genitive singular)
lineaarialgebriesi (genitive plural)
lineaarialgebroidesi (genitive plural)
lineaarialgebroittesi (genitive plural)
lineaarialgebrojesi (genitive plural rare)
lineaarialgebraisi (genitive plural rare)
lineaarialgebraasi (partitive singular)
lineaarialgebriasi (partitive plural)
lineaarialgebroitasi (partitive plural)
lineaarialgebrojasi (partitive plural rare)
lineaarialgebrassasi (inessive singular)
lineaarialgebroissasi (inessive plural)
lineaarialgebrissasi (inessive plural)
lineaarialgebrastasi (elative singular)
lineaarialgebroistasi (elative plural)
lineaarialgebristasi (elative plural)
lineaarialgebraasi (illative singular)
lineaarialgebriisi (illative plural)
lineaarialgebroihisi (illative plural)
lineaarialgebrallasi (adessive singular)
lineaarialgebroillasi (adessive plural)
lineaarialgebrillasi (adessive plural)
lineaarialgebraltasi (ablative singular)
lineaarialgebroiltasi (ablative plural)
lineaarialgebriltasi (ablative plural)
lineaarialgebrallesi (allative singular)
lineaarialgebroillesi (allative plural)
lineaarialgebrillesi (allative plural)
lineaarialgebranasi (essive singular)
lineaarialgebroinasi (essive plural)
lineaarialgebrinasi (essive plural)
lineaarialgebraksesi (singular translative)
lineaarialgebroiksesi (plural translative)
lineaarialgebriksesi (plural translative)
lineaarialgebrattasi (abessive singular)
lineaarialgebroittasi (abessive plural)
lineaarialgebrittasi (abessive plural)
lineaarialgebroinesi (comitative plural)
lineaarialgebrinesi (comitative plural)
lineaarialgebramme (nominative singular)
lineaarialgebramme (nominative plural)
lineaarialgebramme (accusative nominative singular)
lineaarialgebramme (accusative nominative plural)
lineaarialgebramme (accusative genitive singular)
lineaarialgebramme (accusative genitive plural)
lineaarialgebramme (genitive singular)
lineaarialgebriemme (genitive plural)
lineaarialgebroidemme (genitive plural)
lineaarialgebroittemme (genitive plural)
lineaarialgebrojemme (genitive plural rare)
lineaarialgebraimme (genitive plural rare)
lineaarialgebraamme (partitive singular)
lineaarialgebriamme (partitive plural)
lineaarialgebroitamme (partitive plural)
lineaarialgebrojamme (partitive plural rare)
lineaarialgebrassamme (inessive singular)
lineaarialgebroissamme (inessive plural)
lineaarialgebrissamme (inessive plural)
lineaarialgebrastamme (elative singular)
lineaarialgebroistamme (elative plural)
lineaarialgebristamme (elative plural)
lineaarialgebraamme (illative singular)
lineaarialgebriimme (illative plural)
lineaarialgebroihimme (illative plural)
lineaarialgebrallamme (adessive singular)
lineaarialgebroillamme (adessive plural)
lineaarialgebrillamme (adessive plural)
lineaarialgebraltamme (ablative singular)
lineaarialgebroiltamme (ablative plural)
lineaarialgebriltamme (ablative plural)
lineaarialgebrallemme (allative singular)
lineaarialgebroillemme (allative plural)
lineaarialgebrillemme (allative plural)
lineaarialgebranamme (essive singular)
lineaarialgebroinamme (essive plural)
lineaarialgebrinamme (essive plural)
lineaarialgebraksemme (singular translative)
lineaarialgebroiksemme (plural translative)
lineaarialgebriksemme (plural translative)
lineaarialgebrattamme (abessive singular)
lineaarialgebroittamme (abessive plural)
lineaarialgebrittamme (abessive plural)
lineaarialgebroinemme (comitative plural)
lineaarialgebrinemme (comitative plural)
lineaarialgebranne (nominative singular)
lineaarialgebranne (nominative plural)
lineaarialgebranne (accusative nominative singular)
lineaarialgebranne (accusative nominative plural)
lineaarialgebranne (accusative genitive singular)
lineaarialgebranne (accusative genitive plural)
lineaarialgebranne (genitive singular)
lineaarialgebrienne (genitive plural)
lineaarialgebroidenne (genitive plural)
lineaarialgebroittenne (genitive plural)
lineaarialgebrojenne (genitive plural rare)
lineaarialgebrainne (genitive plural rare)
lineaarialgebraanne (partitive singular)
lineaarialgebrianne (partitive plural)
lineaarialgebroitanne (partitive plural)
lineaarialgebrojanne (partitive plural rare)
lineaarialgebrassanne (inessive singular)
lineaarialgebroissanne (inessive plural)
lineaarialgebrissanne (inessive plural)
lineaarialgebrastanne (elative singular)
lineaarialgebroistanne (elative plural)
lineaarialgebristanne (elative plural)
lineaarialgebraanne (illative singular)
lineaarialgebriinne (illative plural)
lineaarialgebroihinne (illative plural)
lineaarialgebrallanne (adessive singular)
lineaarialgebroillanne (adessive plural)
lineaarialgebrillanne (adessive plural)
lineaarialgebraltanne (ablative singular)
lineaarialgebroiltanne (ablative plural)
lineaarialgebriltanne (ablative plural)
lineaarialgebrallenne (allative singular)
lineaarialgebroillenne (allative plural)
lineaarialgebrillenne (allative plural)
lineaarialgebrananne (essive singular)
lineaarialgebroinanne (essive plural)
lineaarialgebrinanne (essive plural)
lineaarialgebraksenne (singular translative)
lineaarialgebroiksenne (plural translative)
lineaarialgebriksenne (plural translative)
lineaarialgebrattanne (abessive singular)
lineaarialgebroittanne (abessive plural)
lineaarialgebrittanne (abessive plural)
lineaarialgebroinenne (comitative plural)
lineaarialgebrinenne (comitative plural)
lineaarialgebransa (nominative singular)
lineaarialgebransa (nominative plural)
lineaarialgebransa (accusative nominative singular)
lineaarialgebransa (accusative nominative plural)
lineaarialgebransa (accusative genitive singular)
lineaarialgebransa (accusative genitive plural)
lineaarialgebransa (genitive singular)
lineaarialgebriensa (genitive plural)
lineaarialgebroidensa (genitive plural)
lineaarialgebroittensa (genitive plural)
lineaarialgebrojensa (genitive plural rare)
lineaarialgebrainsa (genitive plural rare)
lineaarialgebraansa (partitive singular)
lineaarialgebriaan (partitive plural)
lineaarialgebroitaan (partitive plural)
lineaarialgebriansa (partitive plural)
lineaarialgebroitansa (partitive plural)
lineaarialgebrojaan (partitive plural rare)
lineaarialgebrojansa (partitive plural rare)
lineaarialgebrassaan (inessive singular)
lineaarialgebrassansa (inessive singular)
lineaarialgebroissaan (inessive plural)
lineaarialgebrissaan (inessive plural)
lineaarialgebroissansa (inessive plural)
lineaarialgebrissansa (inessive plural)
lineaarialgebrastaan (elative singular)
lineaarialgebrastansa (elative singular)
lineaarialgebroistaan (elative plural)
lineaarialgebristaan (elative plural)
lineaarialgebroistansa (elative plural)
lineaarialgebristansa (elative plural)
lineaarialgebraansa (illative singular)
lineaarialgebriinsa (illative plural)
lineaarialgebroihinsa (illative plural)
lineaarialgebrallaan (adessive singular)
lineaarialgebrallansa (adessive singular)
lineaarialgebroillaan (adessive plural)
lineaarialgebrillaan (adessive plural)
lineaarialgebroillansa (adessive plural)
lineaarialgebrillansa (adessive plural)
lineaarialgebraltaan (ablative singular)
lineaarialgebraltansa (ablative singular)
lineaarialgebroiltaan (ablative plural)
lineaarialgebriltaan (ablative plural)
lineaarialgebroiltansa (ablative plural)
lineaarialgebriltansa (ablative plural)
lineaarialgebralleen (allative singular)
lineaarialgebrallensa (allative singular)
lineaarialgebroilleen (allative plural)
lineaarialgebrilleen (allative plural)
lineaarialgebroillensa (allative plural)
lineaarialgebrillensa (allative plural)
lineaarialgebranaan (essive singular)
lineaarialgebranansa (essive singular)
lineaarialgebroinaan (essive plural)
lineaarialgebrinaan (essive plural)
lineaarialgebroinansa (essive plural)
lineaarialgebrinansa (essive plural)
lineaarialgebrakseen (singular translative)
lineaarialgebraksensa (singular translative)
lineaarialgebroikseen (plural translative)
lineaarialgebrikseen (plural translative)
lineaarialgebroiksensa (plural translative)
lineaarialgebriksensa (plural translative)
lineaarialgebrattaan (abessive singular)
lineaarialgebrattansa (abessive singular)
lineaarialgebroittaan (abessive plural)
lineaarialgebrittaan (abessive plural)
lineaarialgebroittansa (abessive plural)
lineaarialgebrittansa (abessive plural)
lineaarialgebroineen (comitative plural)
lineaarialgebrineen (comitative plural)
lineaarialgebroinensa (comitative plural)
lineaarialgebrinensa (comitative plural)

[ˈline̞ˌɑ̝ːriˌɑ̝lɡe̞brɑ̝]

(mathematics, linear algebra) linear algebra


LU

matrix decomposition
matrix decomposition

(linear algebra) A matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix, used in numerical analysis to solve systems of linear equations or find the inverse of a matrix.


polar cone

Since a is in S and b is in Sᵒ — the polar cone of S — we know that aᵀ b ≤ 0.

(linear algebra) Relative to a set S in Rⁿ, the set of points y such that the inner product of x and y is not positive for any point x in S.


determinante

Determinante on the Spanish Wikipedia.Wikipedia ᵉˢ