In this chapter the emphasis is on the notions of eigenvalue and eigenvector of a square matrix A. These are respectively a scalar λ and a non-zero column matrix x such that Ax = λx In order to compute the λ and the x one begins by considering the system of equations (A - λIn)x = 0. These have a nonzero solution if and only if det (A - λIn) = 0. The left hand side of this equation is a polynomial of degree n which, when made monic, becomes the characteristic polynomial χA(X) of A. The zeros of the characteristic polynomial are thus the eigenvalues. It can be shown that every n × n matrix A satisfies its characteristic polynomial (the Cayley-Hamilton theorem). The minimum polynomial mA(X) of A is the monic polynomial of least degree satisfied by A. It has degree less than or equal to that of χA(A) and divides χA(X).

If the n×n matrix A has n distinct eigenvalues λ1,…,λn and if x1,…,xn are corresponding eigenvectors then the matrix P whose ith column is xi for each i is such that P-1 exists and P-1AP = D where D = [dij] = diag {λ1,…,λn} is the diagonal matrix with dii = λi for every i. When A is real and symmetric, P can be chosen to be orthogonal (P-1 = Pt); this is achieved by normalising the eigenvectors