See Hermitian matrix on Wiktionary
{
"etymology_text": "Named after French mathematician Charles Hermite (1822–1901), who demonstrated in 1855 that such matrices always have real eigenvalues.",
"forms": [
{
"form": "Hermitian matrixes",
"tags": [
"plural"
]
},
{
"form": "Hermitian matrices",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "+",
"2": "Hermitian matrices"
},
"expansion": "Hermitian matrix (plural Hermitian matrixes or Hermitian matrices)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Czech translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Finnish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Icelandic translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Italian translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Polish translations",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Russian translations",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Linear algebra",
"orig": "en:Linear algebra",
"parents": [
"Algebra",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"text": "Hermitian matrices have real diagonal elements as well as real eigenvalues.",
"type": "example"
},
{
"text": "If a Hermitian matrix has a simple spectrum (of eigenvalues) then its eigenvectors are orthogonal.",
"type": "example"
},
{
"text": "If an observable can be described by a Hermitian matrix H, then for a given state #x5C;langleA#x5C;rangle, the expectation value of the observable for that state is #x5C;langleA#x7C;H#x7C;A#x5C;rangle.",
"type": "example"
},
{
"text": "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,\nThere 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)."
},
{
"ref": "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)",
"text": "where H is a hermitian matrix. Now any linear combination of hermitian matrices with real coefficients is again a hermitian matrix."
},
{
"ref": "1998, Eugenio Iannone, Francesco Matera, Antonio Mecozzi, Marina Settembre, Nonlinear Optical Communication Networks, page 442:",
"text": "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.",
"type": "quote"
}
],
"glosses": [
"A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†."
],
"hypernyms": [
{
"word": "normal matrix"
}
],
"hyponyms": [
{
"word": "Pauli matrix"
},
{
"word": "Gramian matrix"
},
{
"word": "self-adjoint matrix"
},
{
"word": "symmetric matrix"
},
{
"word": "real matrix"
}
],
"id": "en-Hermitian_matrix-en-noun-8rb9~Zij",
"links": [
[
"linear algebra",
"linear algebra"
],
[
"square matrix",
"square matrix"
],
[
"complex",
"complex number"
],
[
"conjugate transpose",
"conjugate transpose"
]
],
"raw_glosses": [
"(linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†."
],
"synonyms": [
{
"word": "hermitian matrix"
}
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
],
"translations": [
{
"code": "cs",
"lang": "Czech",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "hermitovská matice"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "square matrix equal to its own conjugate transpose",
"word": "hermiittinen matriisi"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"neuter"
],
"word": "sjálfoka fylki"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"neuter"
],
"word": "hermískt fylki"
},
{
"code": "it",
"lang": "Italian",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "matrice hermitiana"
},
{
"code": "pl",
"lang": "Polish",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "macierz hermitowska"
},
{
"code": "ru",
"lang": "Russian",
"roman": "ermítova mátrica",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "эрми́това ма́трица"
},
{
"code": "ru",
"lang": "Russian",
"roman": "samosoprjažónnaja mátrica",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "самосопряжённая ма́трица"
}
],
"wikipedia": [
"Charles Hermite",
"Hermitian matrix"
]
}
],
"sounds": [
{
"ipa": "/hɝˈmɪ.ʃən ˈmeɪ.tɹɪks/",
"tags": [
"US"
]
}
],
"word": "Hermitian matrix"
}
{
"etymology_text": "Named after French mathematician Charles Hermite (1822–1901), who demonstrated in 1855 that such matrices always have real eigenvalues.",
"forms": [
{
"form": "Hermitian matrixes",
"tags": [
"plural"
]
},
{
"form": "Hermitian matrices",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "+",
"2": "Hermitian matrices"
},
"expansion": "Hermitian matrix (plural Hermitian matrixes or Hermitian matrices)",
"name": "en-noun"
}
],
"hypernyms": [
{
"word": "normal matrix"
}
],
"hyponyms": [
{
"word": "Pauli matrix"
},
{
"word": "Gramian matrix"
},
{
"word": "self-adjoint matrix"
},
{
"word": "symmetric matrix"
},
{
"word": "real matrix"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English eponyms",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with quotations",
"English terms with usage examples",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with entries",
"Terms with Czech translations",
"Terms with Finnish translations",
"Terms with Icelandic translations",
"Terms with Italian translations",
"Terms with Polish translations",
"Terms with Russian translations",
"en:Linear algebra"
],
"examples": [
{
"text": "Hermitian matrices have real diagonal elements as well as real eigenvalues.",
"type": "example"
},
{
"text": "If a Hermitian matrix has a simple spectrum (of eigenvalues) then its eigenvectors are orthogonal.",
"type": "example"
},
{
"text": "If an observable can be described by a Hermitian matrix H, then for a given state #x5C;langleA#x5C;rangle, the expectation value of the observable for that state is #x5C;langleA#x7C;H#x7C;A#x5C;rangle.",
"type": "example"
},
{
"text": "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,\nThere 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)."
},
{
"ref": "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)",
"text": "where H is a hermitian matrix. Now any linear combination of hermitian matrices with real coefficients is again a hermitian matrix."
},
{
"ref": "1998, Eugenio Iannone, Francesco Matera, Antonio Mecozzi, Marina Settembre, Nonlinear Optical Communication Networks, page 442:",
"text": "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.",
"type": "quote"
}
],
"glosses": [
"A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†."
],
"links": [
[
"linear algebra",
"linear algebra"
],
[
"square matrix",
"square matrix"
],
[
"complex",
"complex number"
],
[
"conjugate transpose",
"conjugate transpose"
]
],
"raw_glosses": [
"(linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that A=A^†."
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
],
"wikipedia": [
"Charles Hermite",
"Hermitian matrix"
]
}
],
"sounds": [
{
"ipa": "/hɝˈmɪ.ʃən ˈmeɪ.tɹɪks/",
"tags": [
"US"
]
}
],
"synonyms": [
{
"word": "hermitian matrix"
}
],
"translations": [
{
"code": "cs",
"lang": "Czech",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "hermitovská matice"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "square matrix equal to its own conjugate transpose",
"word": "hermiittinen matriisi"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"neuter"
],
"word": "sjálfoka fylki"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"neuter"
],
"word": "hermískt fylki"
},
{
"code": "it",
"lang": "Italian",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "matrice hermitiana"
},
{
"code": "pl",
"lang": "Polish",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "macierz hermitowska"
},
{
"code": "ru",
"lang": "Russian",
"roman": "ermítova mátrica",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "эрми́това ма́трица"
},
{
"code": "ru",
"lang": "Russian",
"roman": "samosoprjažónnaja mátrica",
"sense": "square matrix equal to its own conjugate transpose",
"tags": [
"feminine"
],
"word": "самосопряжённая ма́трица"
}
],
"word": "Hermitian matrix"
}
Download raw JSONL data for Hermitian matrix meaning in All languages combined (5.0kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.