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.


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


similarity

similarity (countable and uncountable, plural similarities)

From French similarité.
Morphologically similar + -ity

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


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.


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.


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


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


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.


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.


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


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.


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


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.


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


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.


invertible matrix

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


linear combination

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.


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.


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.


antisymmetric

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

The eigenvalues of an antisymmetric matrix are all purely imaginary numbers, and occur as conjugate pairs, #43;iw and -iw. As a corollary it follows that an antisymmetric matrix of odd order necessarily has one eigenvalue equal to zero; antisymmetric matrices of odd order are singular.

1974, Robert McCredie May, Stability and Complexity in Model Ecosystems, Princeton University Press, page 193:

(linear algebra, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof:

(of a matrix) Whose transpose equals its negative (i.e., Mᵀ = −M);

Notice that the tensors defined by:
 #92;textstyleT#95;S#92;equiv#92;frac#123;1#125;#123;2#125;(T#43;Tᵀ), #92;textstyleT#95;A#92;equiv#92;frac#123;1#125;#123;2#125;(T-Tᵀ), (3.47)
are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.

1986, Millard F. Beatty Jr., Principles of Engineering Mechanics, Volume 1: Kinematics - The Geometry of Motion, Plenum Press, page 163:

(of a tensor) That changes sign when any two indices are interchanged (e.g., Tᵢⱼₖ = -Tⱼᵢₖ);

Antisymmetric bilinear forms and wedge products are defined exactly as above, only now they are functions from #92;Rⁿ#92;times#92;Rⁿ to #92;R.[…]
Exercise 21 Show that every antisymmetric bilinear form on #92;R³ is a wedge product of two covectors.

2012, Stephanie Frank Singer, Symmetry in Mechanics: A Gentle, Modern Introduction, Springer, page 28:

(of a bilinear form) For which B(w,v) = -B(v,w).


inner product

(linear algebra) A generalization of the dot product for vectors of any dimensionality that may or may not be complex-numbered.


scalar field

(mathematics, linear algebra) The field (algebraic structure) for which scalar multiplication is defined for a given vector space; field of scalars.


bilinear

The function f(x,y)#61;xy is bilinear, because f(ax#43;bu,y)#61;af(x,y)#43;bf(u,y) and f(x,ay#43;bv)#61;af(x,y)#43;bf(x,v) for all values.

(linear algebra, of a function in two variables) Linear (preserving linear combinations) in each variable.

bilineární

(mathematics) linear in each of two variables


eigenspace

(linear algebra) The linear subspace consisting of all eigenvectors associated with a particular eigenvalue, together with the zero vector.

eigenhapësirë

set of the eigenvectors associated with a particular eigenvalue


eigendecomposition

eigendecompositions (plural)
eigen decomposition (alternative)

(linear algebra) The factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.


contravariance

en-noun

contravariance (countable and uncountable, plural contravariances)

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

(linear algebra) Of vectors, the property of scaling inversely with a change of basis, as opposed to covariance.


antidiagonal

antidiagonals (plural)
anti-diagonal (alternative)

(linear algebra) The diagonal of a matrix that leads from top-right towards bottom-left.

αντιδιαγώνιος

the diagonal of a matrix that leads from top-right towards bottom-left


immanant

(linear algebra) A function or property of a matrix, defined as a generalization of the concepts of determinant and permanent.


nonsingular

(linear algebra, of a matrix) Invertible.


tridiagonal

The following proposition shows a connection between negafibonacci numbers and the determinants of a family of tridiagonal matrices.

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, page 3:

(linear algebra, of a matrix) Having nonzero elements only in the main diagonal and the diagonals directly above and below it.


semisimple

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

(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.

(linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.


scalar multiplication

scalar multiplication (countable and uncountable, plural scalar multiplications)

(linear algebra) Multiplication of a vector by a scalar (which belongs to the vector space's field of scalars).


Cauchy-Schwarz inequality

Named after Augustin-Louis Cauchy and Hermann Amandus Schwarz.

(linear algebra, mathematical analysis) A theorem which states that the absolute value of the dot product between two vectors is less than or equal to the product of the magnitudes of the two vectors.

Augustin-Louis Cauchy
Cauchy-Schwarz inequality
Hermann Amandus Schwarz


Schwarz inequality

Schwarz inequality (plural Schwarz inequalities)

Cauchy-Schwarz inequality
Cauchy-Schwarz inequality

(linear algebra) Cauchy-Schwarz inequality


real vector space

real vector space (plural real vector spaces)

(linear algebra, mathematical analysis) A vector space over the field of real numbers.


convex combination

convex combination (plural convex combinations)

(linear algebra) A linear combination (of vectors in Euclidean space) in which the coefficients are non-negative and all add up to one.


affine combination

affine combination (plural affine combinations)

(linear algebra) A linear combination (of vectors in Euclidean space) in which the coefficients all add up to one.


Hermitian matrix

Hermitian matrix (plural Hermitian matrixes or Hermitian matrices)

Hermitian matrixes (plural)
Hermitian matrices (plural)
hermitian matrix (alternative)

Named after French mathematician Charles Hermite (1822–1901), who demonstrated in 1855 that such matrices always have real eigenvalues.

Hermitian matrices have real diagonal elements as well as real eigenvalues.

If a Hermitian matrix has a simple spectrum (of eigenvalues) then its eigenvectors are orthogonal.

If an observable can be described by a Hermitian matrix H, then for a given state #92;langleA#92;rangle, the expectation value of the observable for that state is #92;langleA#124;H#124;A#92;rangle.

1988, I. M. Gelfand, M. I. Graev, Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry, Israel M. Gelfand, Collected Papers, Volume II, Springer-Verlag, page 366,
There are three types of such spaces: the space of positive definite (or negative definite) Hermitian matrices, the space of nondefinite Hermitian matrices, and finally the space of degenerate Hermitian matrices p, satisfying the condition p ≥ 0 (or p ≤ 0).

where H is a hermitian matrix. Now any linear combination of hermitian matrices with real coefficients is again a hermitian matrix.

1997, A. W. Joshi, Elements of Group Theory for Physicists, New Age International, 4th Edition, page 129, For this we note that if H is a hermitian matrix, exp(iH) is a unitary matrix. The converse is also true, i.e., if U is any unitary matrix, then it can be expressed in the form U = exp(iH), (4.94)

Exploiting the properties of hermitian matrixes [2], it is possible to obtain an analytical expression for the characteristic function of a hermitian quadratic form of gaussian variables, which is useful in the evaluation of transmission system performance.

1998, Eugenio Iannone, Francesco Matera, Antonio Mecozzi, Marina Settembre, Nonlinear Optical Communication Networks, page 442:

(linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†.

hermitovská matice

square matrix equal to its own conjugate transpose


minimal polynomial

minimal polynomial (plural minimal polynomials)

minimipolynomi

monic polynomial of smallest degree for which a given element is a root

Each root of the minimal polynomial of a matrix M is an eigenvalue of M and a root of its characteristic polynomial. (A root of the minimal polynomial has a multiplicity that is less than or equal to the multiplicity of the same root in the characteristic polynomial. Thus the minimal polynomial divides the characteristic polynomial. Also, any root of the characteristic polynomial is also a root of the minimal polynomial, so the two kinds of polynomial have the same roots, only (possibly) differing in their multiplicities.)

1965 [John Wiley], Robert B. Ash, Information Theory, 1990, Dover, page 161,
A procedure for obtaining the minimal polynomial of the matrix Tⁱ, without actually computing the powers of T is indicated in the solution to Problem 5.9.

In this section we will show that if #92;#123;A,B,C,D#92;#125; is a controllable and observable realization of #92;mathbfG, then #92;lambda is a pole of #92;mathbfG if and only if #92;lambda is an eigenvalue of A. Moreover, the roots (multiplicities included) of the minimal polynomial of A are the poles of #92;mathbfG.

2003, Martin J. Corless, Art Frazho, Linear Systems and Control: An Operator Perspective, Marcel Dekker, page 77:

2007, A. R. Vasishta, Vipin Vasishta, A.K. Vasishta, Abstract and Linear Algebra, Krishna Prakashan Media, 3rd Edition, page CA-439,
Theorem 1. The minimal polynomial of a matrix or of a linear operator is unique.

(linear algebra) For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.


eigenpair

(linear algebra) The mathematical pair of an eigenvector (ẕ), and its associated eigenvalue (λ). This is written as (λ,ẕ)


eigenprojection

(linear algebra) The linear operator projecting onto an eigenspace.


column vector

(linear algebra) An n×1 matrix that represents a vector.

列向量

matrix with one column that represents a vector


row vector

(linear algebra) A 1×n matrix that represents a vector.

行向量

matrix with one row that represents a vector


unitary matrix

unitary matrix (plural unitary matrices or unitary matrixes)

unitary matrices (plural)
unitary matrixes (plural)

(linear algebra) A matrix which when multiplied by its conjugate transpose yields the identity matrix.


singular matrix

singular matrix (plural singular matrices or singular matrixes)

singular matrices (plural)
singular matrixes (plural)

(linear algebra) A square matrix which is not invertible.


Hermitian conjugate

Hermitian conjugate (plural Hermitian conjugates)

Named after Charles Hermite (1822–1901), French mathematician.


linear form

In #92;mathbb#123;R#125;ⁿ, if vectors are represented as column vectors, then linear forms are represented as row vectors, and their action on vectors is given by the matrix product, with the row vector on the left and the column vector on the right.

1961 [Prentice-Hall], Richard A. Silverman (translator), Georgi E. Shilov, An Introduction to the Theory of Linear Spaces, 1974, Dover, page 66,
A more general linear form in the same space is the expression
f(x)=∑ₖ₊₁ⁿc_kζₖ
with arbitrary fixed coefficients c_1,c_2…,c_n.

In the theory of distributions, functions are replaced by linear forms on an auxiliary vector space, whose members are called test functions.

1998, F. G. Friedlander, M. Joshi, Introduction to the Theory of Distributions, 2nd edition, Cambridge University Press, page 2:

In order to better understand the effect of reducing a K-vector subspace V#92;subseteqP#95;d modulo a generic linear form, we generalize Proposition 5.5.18 as follows.

2005, Martin Kreuzer, Lorenzo Robbiano, Computational Commutative Algebra 2, Springer, page 264:


bilinear form

(linear algebra) A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to either one of its arguments when the other one is held constant.


orthogonal complement

orthogonal complement (plural orthogonal complements)

(linear algebra, functional analysis) The set of all vectors which are orthogonal to a given set of vectors.


direct sum

(linear algebra) A linear sum in which the intersection of the summands has dimension zero. Equivalently, a linear sum of two subspaces, any vector of which can be expressed uniquely as a sum of two vectors: one vector belonging to the first subspace, and the other vector belonging to the second subspace.

block diagonal matrix
block diagonal matrix

(linear algebra) A block diagonal matrix interpreted as having its non-zero blocks (which are square matrices) as summands.


special unitary group

special unitary group (plural special unitary groups)

1992 [Prentice-Hall], George H. Duffey, Applied Group Theory: For Physicists and Chemists, 2015, Dover, Unabridged Republication, page 284,
The special unitary group in two dimensions is represented by the 2 X 2 unitary matrices whose determinants equal 1.

The group is called the special unitary group in two dimensions, or SU(2), because it acts on matrices of degree 2.

2000, Herbert S. Green, Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process, Springer, page 26:

S³≅SU(2)=Spin 3\2 bigg ↓\SO(3)≅ℝP³

2004, Roger Cooke (translator), Vladimir I. Arnold, Lectures on Partial Differential Equations, [1997, Lekstii ob uravneniyakh s chastnymi proizvodnymi], Springer, page 81, When n = 3, the group of rotations SO(3) is isomorphic to the real three-dimensional projective space ℝP³. It has a two-sheeted covering by the three-dimensional sphere (the group of unit quaternions), which in turn is isomorphic to the special unitary group SU(2), also known as the spin group of order 3, as in the following diagram

(linear algebra, group theory) For given n, the group of n×n unitary matrices with complex elements and determinant equal to one.

erityinen unitaarinen ryhmä

group of n×n matrices with complex elements and determinant 1


left eigenvalue

left eigenvalue (plural left eigenvalues)

(linear algebra) An eigenvalue corresponding to the left transform of a given linear operator; i.e. λ such that urm A=uλ.


right eigenvalue

right eigenvalue (plural right eigenvalues)

(linear algebra) An eigenvalue corresponding to the right transform of a given linear operator; i.e. λ such that rm Au=λu.


proper value

1961 [Oxford University Press], Sterling K. Berberian, Introduction to Hilbert Space, American Mathematical Society (AMS Chelsea), 1999, Reprint, page 178,
In this terminology, Theorem 1 asserts that every non-zero proper value of a CC-operator has finite multiplicity. This result is not always helpful, for there exist CC-operators having no proper values at all:

Thus any proper value of AA* is a proper value of A*A.

1962, A. R. Amir-Moéz, A. L. Fass, Elements of Linear Space, Pergamon Press, page 134:

– the proper values of L;
– for each proper value α of L|Eᶜ and proper value β of L|Eᵘ, the proper value α/β — from the assumptions it follows that this latter collection of proper values consists of contracting proper values only.

2010, F. Takens, A Vanderbauwhede, Local invariant manifolds and normal forms, H. Broer, F. Takens, B. Hasselblatt (editors), Handbook of Dynamical Systems, Volume 3, Elsevier (North-Holland), page 106, The proper values of this linear mapping are


left eigenvector

left eigenvector (plural left eigenvectors)

(linear algebra) An eigenvector corresponding to the left transform of a given linear operator; i.e. u such that urm A=uλ.


right eigenvector

right eigenvector (plural right eigenvectors)

(linear algebra) An eigenvector corresponding to the right transform of a given linear operator; i.e. u such that rm Au=λu.


QR decomposition

QR decomposition (plural QR decompositions)

(linear algebra) A decomposition of a matrix into a product of a unitary matrix (orthogonal matrix) and an upper-triangular matrix.


Toeplitz matrix

Toeplitz matrix (plural Toeplitz matrices)

In [1] two tridiagonal Toeplitz matrices were presented […]

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, Introduction, page 1:

(linear algebra) A matrix in which each descending diagonal from left to right is constant.


Hankel matrix

Hankel matrix (plural Hankel matrices or Hankel matrixes)

Hankel matrices (plural)
Hankel matrixes (plural)

(linear algebra) A square matrix with constant skew-diagonals (positive sloping diagonals).


outer product

(mathematics, linear algebra) The tensor product of two vectors.


REF

row echelon form
row echelon form#English

(mathematics, linear algebra) Initialism of row echelon form.


geometric multiplicity

geometric multiplicity (plural geometric multiplicities)

(linear algebra) The dimension of the eigenspace of a given eigenvalue (of a given square matrix).


Cramer's rule

Named after the Swiss mathematician Gabriel Cramer, who originally published the formula in 1750.

linear equations
linear equation#English

(linear algebra) An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.


symplectic matrix

symplectic matrix (plural symplectic matrices)

Then we define the Krein signature of the eigenvalues of modulus one of a symplectic matrix and use this concept to build the rotation function on the symplectic group.

2001, Alberto Abbondandolo, Morse Theory for Hamiltonian Systems, CRC Press (Chapman & Hall/CRC), page x:

In the previous chapter algorithms for computing the eigenvalues of symplectic matrices have been considered that are based on an elimination process for computing the butterfly form of symplectic matrix. Unfortunately, this approach is not suitable when dealing with large and sparse symplectic matrices as an elimination process can not make full use of the sparsity.

2002, Heike Fassbender, Symplectic Methods for the Symplectic Eigenproblem, Kluwer Academic, page 173:

In order to fully exploit these structures we rely on the properties of Hamiltonian, skew-Hamiltonian, and symplectic matrices discussed in Section 1.5. In Section 4.1 we introduce the elementary symplectic matrices which are used by the structured algorithms.

2012, Dario A. Bini, Bruno Iannazzo, Beatrice Meini, Numerical Solution of Algebraic Riccati Equations, Society for Industrial and Applied Mathematics, page 121:

(linear algebra) For given field F (especially the real numbers), even order 2n and nonsingular skew-symmetric matrix Ω, any 2n×2n matrix M with elements in F such that MᵀΩM = Ω (where Mᵀ denotes the transpose of M).


characteristic root

characteristic root (plural characteristic roots)

Calque of French racine caractéristique, the term preferred by French mathematician Augustin-Louis Cauchy (1789–1857).

1962 [Prentice Hall], Leonard E. Fuller, Basic Matrix Theory, 2017, Dover, page 217,
There is an important relationship between the characteristic roots of a symmetric matrix and the r and p of the associated quadratic form. It is true that the number of positive characteristic roots is p, whereas the number of nonzero characteristic roots is r.

The determinant of a matrix is the product of its characteristic roots.

1985, Takeshi Amemiya, Advanced Econometrics, Harvard University Press, page 459:

2009, Babu Ram, Engineering Mathematics, Pearson Education India, page 13.56,
Theorem 13.54. The characteristic roots of a Hermitian matrix are real.


defective matrix

defective matrix (plural defective matrices)

1994, Sigal Ar, Jin-Yi Cai, Chapter 5: Reliable Benchmarks Using Numerical Stability, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, Society for Industrial and Applied Mathematics, page 41,
An n x n matrix A is a defective matrix if it has a defective eigenvalue.
Obviously, a Jordan block of dimension greater than 1, and a matrix whose Jordan canonical form has a Jordan block of dimension greater than 1, are defective matrices.

Therefore, the sum of the geometric multiplicities of a defective matrix will be less than n.

2007, Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Springer, page 261:

Thus, defective matrices exhibit high sensitivity to imperfections.
It can be checked that the unperturbed matrices L* are in fact nearly-defective, because they have nearly-parallel eigenvectors. In particular, the matrices are themselves perturbations of order ε of exactly defective matrices L_(id).

2013, Angelo Luongo, “Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures”, in Alexander F. Vakakis, editor, Normal Modes and Localization in Nonlinear Systems, Springer, page 153:

(linear algebra) A square (n×n) matrix that has fewer than n linearly independent eigenvectors, and is therefore not diagonalisable.


symplectic group

symplectic group (plural symplectic groups)

So named by German mathematician Hermann Weyl, replacing previous confusing names. More at symplectic.

This representation ω arises by virtue of the existence of an action by automorphisms of the symplectic group on a certain two-step nilpotent group, commonly called the Heisenberg group.
Weil was of course working in the adelic formalism of modern number theory, so he considered symplectic groups and Heisenberg groups with coefficients in a general local field.

1985, Roger Howe, “Dual Pairs in Physics: Harmonic Oscillators, Photons, Electrons, and Singletons”, in Mosh Flato, Paul Sally, Gregg Zuckerman, editors, Applications of Group Theory in Physics and Mathematical Physics, American Mathematical Society, page 179:

The exponential of an operator gives the exponential mapping #92;textstyleH#92;mapsto#92;text#123;exp#125;(H)#61;#92;sum#123;Hᵏ#47;k#33;#125; of the space of Hamiltonian operators to the symplectic group. The symplectic group acts by conjugation on itself and on its Lie algebra.

2001, V. I. Arnol'd, A. B. Givental', “Symplectic Geometry”, in G. Wassermann, transl., edited by V. I. Arnol'd and S. P. Novikov, Dynamical Systems IV: Symplectic Geometry and its Applications, 2nd edition, Springer, page 18:

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group.

2012, Rolf Berndt, Ralf Schmidt, Elements of the Representation Theory of the Jacobi Group, Springer, page v:

(linear algebra, group theory) For given field F and positive integer n, the group of 2n×2n symplectic matrices with elements in F.


orthogonal matrix

orthogonal matrix (plural orthogonal matrices)

(linear algebra) A square matrix whose columns, considered as vectors, are orthonormal to each other. This implies that the transpose of such a matrix is also its inverse.


transformation matrix

transformation matrix (plural transformation matrices or transformation matrixes)

transformation matrices (plural)
transformation matrixes (plural)

Given a linear transformation T(x) in functional form, its transformation matrix can be constructed by applying T to each vector of the standard basis, then inserting the results into the columns of the new matrix.

A transformation matrix of dimension n×m operates on a column vector of dimension m×1 to produce a row vector of dimension 1×n.

1963 [McGraw-Hill], Lawrence P. Huelsman, Circuits, Matrices and Linear Vector Spaces, 2011, Dover, page 191,
We would like to make as many as possible of the elements of the transformation matrix equal zero.

1968 [McGraw-Hill], Granino A. Korn, Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, 2000, Dover, page 414,
Refer to Sec. 14.8-6 for a procedure yielding transformation matrices T with the desired properties.

The 4x4 transformation matrices are conveniently used to convert various entities expressed in different coordinate systems into another.

2005, Gerard Kim, Designing Virtual Reality Systems: The Structured Approach, Volume 1, Springer, page 47:

(linear algebra) A matrix (of dimension n×m) that represents some linear transformation from ℝᵐ→ℝⁿ.

kuvausmatriisi

matrix that represents a linear transformation


offdiagonal

(linear algebra) That is not an element on the diagonal of a square matrix.


projection matrix

projection matrix (plural projection matrices)

2006, H. Watari, Y. Shimoyama, Chapter 5: Density Matrix Formalism of Angular Momentum in Multi-Quantum Magnetic Resonance, Christopher J. Bender, Lawrence Berliner (editors, Computational and Instrumental Methods in EPR, Springer, page 191,
The resultant eigenvalues of the angular momentum are independent of direction, but the projection matrices are dependent.

We use the GLU utility function gluPerspective() to set the projection matrix and define the whole image to be the view port (essentially what is to be shown).

2012, Jan Erik Solem, Programming Computer Vision with Python, O'Reilly, page 91:

(linear algebra) A transformation matrix whose associated transformation is a projection.


skew-Hermitian

skew hermitian (alternative)
skew-hermitian (alternative)
skew Hermitian (alternative)

(linear algebra) Of a matrix T, such that T^†=-T.


back substitution

back substitution (countable and uncountable, plural back substitutions)

(linear algebra) A method of solving linear systems that have been transformed into row echelon form.


Gaussian elimination

Gaussian elimination (countable and uncountable, plural Gaussian eliminations)

1976, Thomas E. Rosmond, Frank D. Faulkner, Direct Solution of Elliptic Equations by Block Cyclic Reduction and Factorization, Monthly Weather Review, Volume 104, Issues 1-6, American Meteorological Society, page 644,
We then repeat the Gaussian elimination procedure for the ( boldsymbol E-α₂⁽²⁾) factor in (35), obtaining another intermediate solution vector. It is immediately apparent in this example that four Gaussian eliminations are necessary to obtain ψ_(i,4).

Theoretical partitioning and resource allocation for the Gaussian elimination method used to solve the matrix are discussed.

1989, Richard A. Blech, “Parallel Gaussian Elimination of a Block Tridiagonal Matrix Using Multiple Microcomputers”, in Technical Report, NASA, page 1:

The use of Gaussian eliminations limits the range of functionality of the algorithm, which operates only with square matrices A.

1996, E. D'Hollander, G. R. Joubert, F. J. Peters, D. Trystram, Parallel Computing: State-of-the-art and Perspectives, Elsevier, page 642:

(linear algebra) A method of reducing an augmented matrix to row echelon form.


Gauss-Jordan elimination

Gauss-Jordan elimination (plural Gauss-Jordan eliminations)

(linear algebra) A method of reducing an augmented matrix to reduced row echelon form.


row echelon form

row echelon form (countable and uncountable, plural row echelon forms)

(linear algebra) The stepped appearance of a matrix that has 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.


reduced row echelon form

reduced row echelon form (plural reduced row echelon forms)

(linear algebra) The form of a row-reduced matrix which is in row echelon form.


augmented matrix

augmented matrix (plural augmented matrices)

(linear algebra) Matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.


coefficient matrix

coefficient matrix (plural coefficient matrices)

(linear algebra) A matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.


linear span

(linear algebra) A linear subspace generated by all linear combinations of a given subset of vectors of a given vector space.


linear subspace

linear subspace (plural linear subspaces)

(linear algebra) A subset of vectors of a vector space which is closed under the addition and scalar multiplication of that vector space.


linear sum

(linear algebra) A linear subspace which is the linear span of the union of two subspaces (of some vector space).


rank-nullity theorem

If — for a homogeneous system of linear equations — there are V unknowns and R (linearly independent) equations then, according to the rank-nullity theorem, the solution space is N equals V − R dimensional.

(linear algebra) A theorem about linear transformations (or the matrices that represent them) stating that the rank plus the nullity equals the dimension of the entire vector space (which is the linear transformation’s domain).


row-reduced

(linear algebra) Said of a matrix which has undergone row reduction; i.e., of a matrix which has a 1 as the leading entry of each one of its non-zero rows and which has zeroes as all the other entries of any column containing such a leading entry (of any non-zero row).


column echelon form

column echelon form (plural column echelon forms)

(linear algebra) The transpose of the row echelon form of the transpose of a given matrix.


left nullspace

(linear algebra) The vector space of all row vectors whose dot products with the columns of a given matrix are zero; the nullspace of the transpose of a given matrix.


elementary matrix

elementary matrix (plural elementary matrices)

When an elementary matrix multiplies a square matrix from the left, it applies to that matrix an elementary row operation homologous to the one that created its own self. (Likewise, when an elementary matrix multiplies a square matrix from the right, it applies to that matrix an elementary column operation homologous to the one that created its own self.)

If a series of elementary matrices applied to a given square matrix row-reduce it to the identity matrix, then the same series of elementary matrices applied to the identity matrix yield the inverse of the given matrix.

(linear algebra) A square matrix which is obtained from the identity matrix by a single elementary row (or column) operation.


characteristic vector

characteristic vector (plural characteristic vectors)


positive-definite

(linear algebra) Said of a real, square matrix: that the product of it with a column vector on its right side and the transpose of that column vector on its left side is greater or equal to zero, only equaling zero if the vector itself is zero.

(linear algebra) Said of a quadratic form: that the application of it to a tuple is greater or equal to zero, only equaling zero if the tuple itself is zero.


cross term

(linear algebra) In a quadratic form, a term which is not a squared variable; i.e., a term with more than one variable.


anti-diagonal

(linear algebra) Alternative spelling of antidiagonal.


field of scalars

field of scalars (plural fields of scalars)


multilinear form

multilinear form (plural multilinear forms)

forme multilinéaire

multiply linear mapping to a field of scalars

1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163,
Finally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (and Hankel operators too) really should be viewed as forms, not operators.

1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454,
A multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate m-multilinear form can more concisely be written as M!f.

Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).

2003, Maks A. Akivis, Vladislav V. Goldberg, translated by Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 55:

(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;

Given a vector space V over a field K of scalars, a mapping Vᵏ → K that is linear in each of its arguments;
(more generally) a similarly multiply linear mapping Mʳ → R defined for a given module M over some commutative ring R.


numerical linear algebra

(computing) The study of numerical analysis algorithms to solve mathematical problems in the field of linear algebra.

numerical linear algebra
simple:numerical linear algebra


null matrix

null matrix (plural null matrices or null matrixes)

null matrices (plural)
null matrixes (plural)


trivial solution

trivial solution (plural trivial solutions)

(linear algebra) the answer ⃑x=⃑0 to a linear system A⃑x=⃑0


nontrivial solution

nontrivial solution (plural nontrivial solutions)

(linear algebra) any other answers apart from ⃑x=⃑0 to the linear system A⃑x=⃑0, signifying linear dependence of the linear system


SVD

SVD (countable and uncountable, plural SVDs)

singular value decomposition
singular value decomposition#English

(linear algebra) Initialism of singular value decomposition.


rank-deficient

en-adj

rank-deficient (comparative more rank-deficient, superlative most rank-deficient)

more rank-deficient (comparative)
most rank-deficient (superlative)

(mathematics, linear algebra, of a matrix) Not having full rank; having a rank (dimension) less than the number of columns and the number of rows.


full rank

full rank (comparative more full rank, superlative most full rank)

more full rank (comparative)
most full rank (superlative)

(mathematics, linear algebra, of a matrix) Having the highest possible rank for its size; having a rank (dimension) equal to the number of columns or the number of rows.


ladder operator

en-noun

ladder operator (plural ladder operators)

(linear algebra) An operator that either increases or decreases the eigenvalue of another operator.


idemfactor

idemfactor (usually uncountable, plural idemfactors)

(mathematics, linear algebra) An operator or quantity which is at both idemfacient and idemfaciend.


idemfacient

(mathematics, linear algebra) When multiplying an idempotent matrix produces itself.


idemfaciend

The possibility of the selection of all the remaining units as idemfaciend or nilfaciend is easily established.

1882, Benjamin Peirce, Linear Associative Algebra, page 14:

(mathematics, linear algebra) When multiplied by an idempotent matrix produces itself.


Schur decomposition

Named after Russian mathematician Issai Schur.

named-after

(linear algebra) A matrix decomposition allowing one to write complex square matrices as similar to a triangular matrix whose diagonal elements are the eigenvalues of the original matrix.


basis vector

(linear algebra) A vector that forms part of a basis for a vector space.


skew-symmetric matrix

en-noun

skew-symmetric matrix (plural skew-symmetric matrices or skew-symmetric matrixes)

skew-symmetric matrices (plural)
skew-symmetric matrixes (plural)

(linear algebra) A square matrix equal to the negative of its transpose.


skew-symmetric determinant

skew-symmetric determinant (plural skew-symmetric determinants)

skew-symmetric matrix
skew-symmetric matrix

(linear algebra) The determinant of a skew-symmetric matrix.


triangular matrix

triangular matrix (plural triangular matrixes or triangular matrices)

triangular matrixes (plural)
triangular matrices (plural)

(linear algebra) A square matrix in which all the items on one side of the main diagonal are zero.