@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr:-JR0BZJDR-C skos:prefLabel "square matrix"@en, "matrice carrée"@fr ; a skos:Concept ; skos:narrower psr:-P43W3M96-V . psr: a skos:ConceptScheme . psr:-RF2X9JGV-G skos:prefLabel "endomorphism"@en, "endomorphisme"@fr ; a skos:Concept ; skos:narrower psr:-P43W3M96-V . psr:-P43W3M96-V skos:exactMatch , ; skos:definition """En algèbre linéaire, le théorème de Cayley-Hamilton affirme que tout endomorphisme d'un espace vectoriel de dimension finie sur un corps commutatif quelconque annule son propre polynôme caractéristique. En termes de matrice, cela signifie que si A est une matrice carrée d'ordre n et si

{\\\\displaystyle p(X)=\\\\det(XI_{n}-A)=X^{n}+p_{n-1}X^{n-1}+\\\\ldots +p_{1}X+p_{0}}

est son polynôme caractéristique (polynôme d'indéterminée X), alors en remplaçant formellement X par la matrice A dans le polynôme, le résultat est la matrice nulle :

{\\\\displaystyle p(A)=A^{n}+p_{n-1}A^{n-1}+\\\\ldots +p_{1}A+p_{0}I_{n}=0_{n}.\\\\;}

Le théorème de Cayley-Hamilton s'applique aussi à des matrices carrées à coefficients dans un anneau commutatif quelconque. Un corollaire important du théorème de Cayley-Hamilton affirme que le polynôme minimal d'une matrice donnée est un diviseur de son polynôme caractéristique. Bien qu'il porte les noms des mathématiciens Arthur Cayley et William Hamilton, la première démonstration du théorème est donnée par Ferdinand Georg Frobenius en 1878, Cayley l'ayant principalement utilisé dans ses travaux, et Hamilton l'ayant démontré en dimension 2.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Cayley-Hamilton)"""@fr, """In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation. The characteristic polynomial of an

n \times n

matrix

A

is defined as

{\\\\displaystyle p_{A}(\\\\lambda )=\\\\det(\\\\lambda I_{n}-A)}

, where

det

is the determinant operation,

λ

is a variable scalar element of the base ring, and

I n

is the

n \times n

identity matrix. Since each entry of the matrix

{\\\\displaystyle (\\\\lambda I_{n}-A)}

is either constant or linear in

λ

, the determinant of

{\\\\displaystyle (\\\\lambda I_{n}-A)}

is a degree-

n

monic polynomial in

λ

, so it can be written as

{\\\\displaystyle p_{A}(\\\\lambda )=\\\\lambda ^{n}+c_{n-1}\\\\lambda ^{n-1}+\\\\cdots +c_{1}\\\\lambda +c_{0}.}

By replacing the scalar variable

λ

with the matrix

A

, one can define an analogous matrix polynomial expression,

{\\\\displaystyle p_{A}(A)=A^{n}+c_{n-1}A^{n-1}+\\\\cdots +c_{1}A+c_{0}I_{n}.}

(Here,

{\\\\displaystyle A}

is the given matrix—not a variable, unlike

{\\\\displaystyle \\\\lambda }

—so

{\\\\displaystyle p_{A}(A)}

is a constant rather than a function.) The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that

{\\\\displaystyle p_{A}(A)=\\\\mathbf {0} ;}

that is, the characteristic polynomial

{\\\\displaystyle p_{A}}

is an annihilating polynomial for

{\\\\displaystyle A.}

One use for the Cayley–Hamilton theorem is that it allows

A

^$n$ to be expressed as a linear combination of the lower matrix powers of

A

{\\\\displaystyle A^{n}=-c_{n-1}A^{n-1}-\\\\cdots -c_{1}A-c_{0}I_{n}.}

When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions. This corresponds to the special case of certain

4 \times 4

real or

2 \times 2

complex matrices. Cayley in 1858 stated the result for

3 \times 3

and smaller matrices, but only published a proof for the

2 \times 2

case. As for

n \times n

matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”. The general case was first proved by Ferdinand Frobenius in 1878.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem)"""@en ; skos:broader psr:-JR0BZJDR-C, psr:-RF2X9JGV-G ; skos:prefLabel "théorème de Cayley-Hamilton"@fr, "Cayley-Hamilton theorem"@en ; a skos:Concept ; dc:modified "2024-10-18"^^xsd:date ; skos:inScheme psr: .