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 de niteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder re ectors. Two algorithms are proposed, one using non-orthogonal rank-one modi cations of the identity matrix and the other, more costly but more stable, using a combination of Householder re ectors and non-orthogonal rank-one modi cations 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