Matrix (mathematics) - Square Matrices - Main Operations - Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

A number λ and a non-zero vector v satisfying

Av = λv

are called an eigenvalue and an eigenvector of A, respectively. The number λ is an eigenvalue of an n×n-matrix A if and only if A−λI_n is not invertible, which is equivalent to

The polynomial p_A in an indeterminate X given by evaluation the determinant det(XI_n−A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation p_A(λ) = 0 has at most n different solutions, i.e., eigenvalues of the matrix. They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, p_A(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.