Characteristic polynomial

In linear algebra the characteristic polynomial of a square matrix is a polynomial which has the eigenvalues of the matrix as roots.

Let A be an n×n matrix. The characteristic polynomial of A is the determinant

$\chi _{A}(X)=\det(A-XI_{n}),\,$

where X is an indeterminate and I_n is an identity matrix.

The characteristic polynomial is unchanged under similarity, and hence be defined for an endomorphism of a vector space, independent of choice of basis.

Properties

• The characteristic polynomial is monic of degree n;
• The set of roots of the characteristic polynomial is equal to the set of eigenvalues of A.

Cayley-Hamilton theorem

The Cayley-Hamilton theorem states that a matrix satisfies its own characteristic polynomial.