An analysis of the HR algorithm for computing the eigenvalues of a matrix

The HR algorithm, a method of computing the eigenvalues of a matrix, is presented. It is based on the fact that almost every complex square matrix A can be decomposed into a product A = HR of a so-called pseudo-Hermitian matrix H and an upper triangular matrix R. This algorithm is easily seen to be a generalization of the well-known QR algorithm. It is shown how it is related to the power method and inverse iteration, and for special matrices the connection between the LR and HR algorithms is indicated.