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 In is an identity matrix.

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.