This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of n × n matrices.

Given the n × n matrices A and B, the generalized eigenvalue problem seeks the eigenpairs (lambda_k, x_k), solutions of the equation Ax = lambda Bx, or (A - lambda B) x = 0. If B is nonsingular, the eigenpairs of B^-1 A are solutions. If a matrix S exists for which S^T A S = Lambda, and S^T B S = I, where Lambda is a diagonal matrix and I is the n × n identity, then A and B are said to be diagonalized simultaneously, in which case the diagonal entries of Lambda are the generalized eigenvalues for A and B. Such a matrix S exists if A is symmetric and B is positive definite. (Our definition of positive definite includes symmetry.)