: Matemáticas: Cayley-Hamilton theorem

algebra > linear algebra > matrix > square matrix > Cayley-Hamilton theorem

category theory > morphism > endomorphism > Cayley-Hamilton theorem

algebra > abstract algebra > morphism > endomorphism > Cayley-Hamilton theorem

Término preferido

Cayley-Hamilton theorem

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)