Cayley?Hamilton theorem

In linear algebra, the Cayley?Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation.

Related Topics:
Linear algebra - Arthur Cayley - William Hamilton - Square matrix - Real - Complex - Field

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This means the following: if A is the given square n×n matrix and In  is the n×n

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

identity matrix, then the characteristic polynomial of A is defined as:

Related Topics:
Identity matrix - Characteristic polynomial

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:p(t)=det(A-tI_n),

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where "det" is the determinant function. The Cayley?Hamilton theorem states that replacing t by the matrix A in

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

the characteristic polynomial results in the zero matrix:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:p(A)=0.,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Indeed, the Cayley?Hamilton theorem holds for square matrices over commutative rings as well.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An important corollary of the Cayley?Hamilton theorem is that the minimal polynomial of a given matrix is a divisor of its characteristic polynomial. This is very useful in finding the Jordan form of a matrix.

Related Topics:
Corollary - Minimal polynomial - Divisor - Characteristic polynomial - Jordan form

~ ~ ~ ~ ~ ~ ~ ~ ~ ~