Simultaneous tridiagonalization of two symmetric matrices

We show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non-singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non-orthogonal rank-one modifications of the identity matrix and the othe, more costly but more stable, using a combination of Householder reflectors and non-orthogonal rank-one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n3) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments. © 2003 John Wiley and Sons, Ltd.