See characteristic polynomial in All languages combined, or Wiktionary
{
"forms": [
{
"form": "characteristic polynomials",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "characteristic polynomial (plural characteristic polynomials)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"_dis1": "53 47",
"word": "characteristic equation"
},
{
"_dis1": "53 47",
"word": "characteristic root"
},
{
"_dis1": "53 47",
"word": "characteristic vector"
},
{
"_dis1": "53 47",
"word": "eigenvalue"
},
{
"_dis1": "53 47",
"word": "eigenvector"
},
{
"_dis1": "53 47",
"word": "minimal polynomial"
}
],
"senses": [
{
"categories": [
{
"kind": "other",
"langcode": "en",
"name": "Linear algebra",
"orig": "en:Linear algebra",
"parents": [],
"source": "w"
}
],
"examples": [
{
"bold_text_offsets": [
[
4,
29
]
],
"text": "The characteristic polynomial of #92;textstyle#92;left(#92;begin#123;array#125;#123;cc#125;1amp;4#92;#92;3amp;-5#92;end#123;array#125;#92;right) is #92;textstyle#92;left#92;vert#92;begin#123;array#125;#123;cc#125;1-xamp;4#92;#92;3amp;-5-x#92;end#123;array#125;#92;right#92;vert#61;x²#43;4x-17.",
"type": "example"
},
{
"bold_text_offsets": [
[
4,
29
]
],
"text": "The characteristic polynomial of a 2#92;times 2 matrix M is #92;lambda²-#92;mbox#123;tr#125;(M)#92;lambda#43;#92;mbox#123;det#125;(M), where #92;mbox#123;tr#125;(M) denotes the trace of M and #92;mbox#123;det#125;(M) denotes the determinant of M.",
"type": "example"
},
{
"bold_text_offsets": [
[
4,
29
]
],
"text": "The characteristic polynomial of a 3#92;times 3 matrix M is -#92;lambda³#43;#92;mbox#123;tr#125;(M)#92;lambda²-#92;mbox#123;tr#125;(#92;mbox#123;adj#125;(M))#92;lambda#43;#92;mbox#123;det#125;(M), where #92;mbox#123;adj#125;(M) denotes the adjugate of M.",
"type": "example"
}
],
"glosses": [
"The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant."
],
"id": "en-characteristic_polynomial-en-noun-aJsN4Lc6",
"links": [
[
"linear algebra",
"linear algebra"
],
[
"polynomial",
"polynomial"
],
[
"square matrix",
"square matrix"
],
[
"identity matrix",
"identity matrix"
],
[
"indeterminant",
"indeterminant"
],
[
"determinant",
"determinant"
]
],
"raw_glosses": [
"(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."
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
],
"translations": [
{
"_dis1": "68 32",
"code": "ast",
"lang": "Asturian",
"lang_code": "ast",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polinomiu característicu"
},
{
"_dis1": "68 32",
"code": "cmn",
"lang": "Chinese Mandarin",
"lang_code": "cmn",
"sense": "linear algebra",
"word": "特徵多項式 /特征多项式"
},
{
"_dis1": "68 32",
"code": "et",
"lang": "Estonian",
"lang_code": "et",
"sense": "linear algebra",
"word": "karakteristlik polünoom"
},
{
"_dis1": "68 32",
"code": "fi",
"lang": "Finnish",
"lang_code": "fi",
"sense": "linear algebra",
"word": "karakteristinen polynomi"
},
{
"_dis1": "68 32",
"code": "fr",
"lang": "French",
"lang_code": "fr",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polynôme caractéristique"
},
{
"_dis1": "68 32",
"code": "de",
"lang": "German",
"lang_code": "de",
"sense": "linear algebra",
"tags": [
"neuter"
],
"word": "charakteristiches Polynom"
},
{
"_dis1": "68 32",
"code": "hu",
"lang": "Hungarian",
"lang_code": "hu",
"sense": "linear algebra",
"word": "karakterisztikus polinom"
},
{
"_dis1": "68 32",
"code": "it",
"lang": "Italian",
"lang_code": "it",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polinomio caratteristico"
},
{
"_dis1": "68 32",
"code": "pl",
"lang": "Polish",
"lang_code": "pl",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "wielomian charakterystyczny"
},
{
"_dis1": "68 32",
"code": "pt",
"lang": "Portuguese",
"lang_code": "pt",
"sense": "linear algebra",
"tags": [
"Portugal",
"masculine"
],
"word": "polinómio característico"
},
{
"_dis1": "68 32",
"code": "pt",
"lang": "Portuguese",
"lang_code": "pt",
"sense": "linear algebra",
"tags": [
"Brazil",
"masculine"
],
"word": "polinômio característico"
},
{
"_dis1": "68 32",
"code": "ro",
"lang": "Romanian",
"lang_code": "ro",
"sense": "linear algebra",
"tags": [
"neuter"
],
"word": "polinom caracteristic"
},
{
"_dis1": "68 32",
"code": "es",
"lang": "Spanish",
"lang_code": "es",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polinomio característico"
},
{
"_dis1": "68 32",
"code": "sv",
"lang": "Swedish",
"lang_code": "sv",
"sense": "linear algebra",
"tags": [
"neuter"
],
"word": "karakteristiskt polynom"
}
]
},
{
"categories": [
{
"kind": "other",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [],
"source": "w"
},
{
"_dis": "33 67",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "17 83",
"kind": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "28 72",
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "25 75",
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "31 69",
"kind": "other",
"name": "Terms with Asturian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "26 74",
"kind": "other",
"name": "Terms with Estonian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "24 76",
"kind": "other",
"name": "Terms with Finnish translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "24 76",
"kind": "other",
"name": "Terms with French translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "24 76",
"kind": "other",
"name": "Terms with German translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "33 67",
"kind": "other",
"name": "Terms with Hungarian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "22 78",
"kind": "other",
"name": "Terms with Italian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "30 70",
"kind": "other",
"name": "Terms with Mandarin translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "26 74",
"kind": "other",
"name": "Terms with Polish translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "23 77",
"kind": "other",
"name": "Terms with Portuguese translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "26 74",
"kind": "other",
"name": "Terms with Romanian translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "19 81",
"kind": "other",
"name": "Terms with Spanish translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "26 74",
"kind": "other",
"name": "Terms with Swedish translations",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "32 68",
"kind": "other",
"langcode": "en",
"name": "Algebra",
"orig": "en:Algebra",
"parents": [],
"source": "w+disamb"
},
{
"_dis": "33 67",
"kind": "other",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [],
"source": "w+disamb"
}
],
"glosses": [
"A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t)."
],
"id": "en-characteristic_polynomial-en-noun-kT7KyV0v",
"links": [
[
"mathematics",
"mathematics"
],
[
"homogeneous",
"homogeneous"
],
[
"linear",
"linear"
],
[
"ordinary differential equation",
"ordinary differential equation"
],
[
"differential operator",
"differential operator"
]
],
"raw_glosses": [
"(mathematics) A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t)."
],
"topics": [
"mathematics",
"sciences"
]
}
],
"word": "characteristic polynomial"
}
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with entries",
"Terms with Asturian translations",
"Terms with Estonian translations",
"Terms with Finnish translations",
"Terms with French translations",
"Terms with German translations",
"Terms with Hungarian translations",
"Terms with Italian translations",
"Terms with Mandarin translations",
"Terms with Polish translations",
"Terms with Portuguese translations",
"Terms with Romanian translations",
"Terms with Spanish translations",
"Terms with Swedish translations",
"en:Algebra",
"en:Mathematics"
],
"forms": [
{
"form": "characteristic polynomials",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "characteristic polynomial (plural characteristic polynomials)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "characteristic equation"
},
{
"word": "characteristic root"
},
{
"word": "characteristic vector"
},
{
"word": "eigenvalue"
},
{
"word": "eigenvector"
},
{
"word": "minimal polynomial"
}
],
"senses": [
{
"categories": [
"English terms with usage examples",
"en:Linear algebra"
],
"examples": [
{
"bold_text_offsets": [
[
4,
29
]
],
"text": "The characteristic polynomial of #92;textstyle#92;left(#92;begin#123;array#125;#123;cc#125;1amp;4#92;#92;3amp;-5#92;end#123;array#125;#92;right) is #92;textstyle#92;left#92;vert#92;begin#123;array#125;#123;cc#125;1-xamp;4#92;#92;3amp;-5-x#92;end#123;array#125;#92;right#92;vert#61;x²#43;4x-17.",
"type": "example"
},
{
"bold_text_offsets": [
[
4,
29
]
],
"text": "The characteristic polynomial of a 2#92;times 2 matrix M is #92;lambda²-#92;mbox#123;tr#125;(M)#92;lambda#43;#92;mbox#123;det#125;(M), where #92;mbox#123;tr#125;(M) denotes the trace of M and #92;mbox#123;det#125;(M) denotes the determinant of M.",
"type": "example"
},
{
"bold_text_offsets": [
[
4,
29
]
],
"text": "The characteristic polynomial of a 3#92;times 3 matrix M is -#92;lambda³#43;#92;mbox#123;tr#125;(M)#92;lambda²-#92;mbox#123;tr#125;(#92;mbox#123;adj#125;(M))#92;lambda#43;#92;mbox#123;det#125;(M), where #92;mbox#123;adj#125;(M) denotes the adjugate of M.",
"type": "example"
}
],
"glosses": [
"The polynomial produced from a given square matrix by first subtracting the appropriate identity matrix multiplied by an indeterminant and then calculating the determinant."
],
"links": [
[
"linear algebra",
"linear algebra"
],
[
"polynomial",
"polynomial"
],
[
"square matrix",
"square matrix"
],
[
"identity matrix",
"identity matrix"
],
[
"indeterminant",
"indeterminant"
],
[
"determinant",
"determinant"
]
],
"raw_glosses": [
"(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."
],
"topics": [
"linear-algebra",
"mathematics",
"sciences"
]
},
{
"categories": [
"en:Mathematics"
],
"glosses": [
"A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t)."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"homogeneous",
"homogeneous"
],
[
"linear",
"linear"
],
[
"ordinary differential equation",
"ordinary differential equation"
],
[
"differential operator",
"differential operator"
]
],
"raw_glosses": [
"(mathematics) A polynomial P(r) corresponding to a homogeneous, linear, ordinary differential equation P(D) y = 0 where D is a differential operator (with respect to a variable t, if y is a function of t)."
],
"topics": [
"mathematics",
"sciences"
]
}
],
"translations": [
{
"code": "ast",
"lang": "Asturian",
"lang_code": "ast",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polinomiu característicu"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"lang_code": "cmn",
"sense": "linear algebra",
"word": "特徵多項式 /特征多项式"
},
{
"code": "et",
"lang": "Estonian",
"lang_code": "et",
"sense": "linear algebra",
"word": "karakteristlik polünoom"
},
{
"code": "fi",
"lang": "Finnish",
"lang_code": "fi",
"sense": "linear algebra",
"word": "karakteristinen polynomi"
},
{
"code": "fr",
"lang": "French",
"lang_code": "fr",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polynôme caractéristique"
},
{
"code": "de",
"lang": "German",
"lang_code": "de",
"sense": "linear algebra",
"tags": [
"neuter"
],
"word": "charakteristiches Polynom"
},
{
"code": "hu",
"lang": "Hungarian",
"lang_code": "hu",
"sense": "linear algebra",
"word": "karakterisztikus polinom"
},
{
"code": "it",
"lang": "Italian",
"lang_code": "it",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polinomio caratteristico"
},
{
"code": "pl",
"lang": "Polish",
"lang_code": "pl",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "wielomian charakterystyczny"
},
{
"code": "pt",
"lang": "Portuguese",
"lang_code": "pt",
"sense": "linear algebra",
"tags": [
"Portugal",
"masculine"
],
"word": "polinómio característico"
},
{
"code": "pt",
"lang": "Portuguese",
"lang_code": "pt",
"sense": "linear algebra",
"tags": [
"Brazil",
"masculine"
],
"word": "polinômio característico"
},
{
"code": "ro",
"lang": "Romanian",
"lang_code": "ro",
"sense": "linear algebra",
"tags": [
"neuter"
],
"word": "polinom caracteristic"
},
{
"code": "es",
"lang": "Spanish",
"lang_code": "es",
"sense": "linear algebra",
"tags": [
"masculine"
],
"word": "polinomio característico"
},
{
"code": "sv",
"lang": "Swedish",
"lang_code": "sv",
"sense": "linear algebra",
"tags": [
"neuter"
],
"word": "karakteristiskt polynom"
}
],
"word": "characteristic polynomial"
}
Download raw JSONL data for characteristic polynomial meaning in English (5.5kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2026-03-25 from the enwiktionary dump dated 2026-03-03 using wiktextract (05c257f and 9d9a410). 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.