On the simultaneous tridiagonalization of two symmetric matrices

Abstract

We discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal-tridiagonal form under the very mild assumption that the matrix pencil is regular. We outline the general principles and propose a unified framework for the problem. This allows us to gain new insights, leading to an economical approach that only uses Gauss transformations and orthogonal Householder transformations. Numerical experiments show that the approach is numerically robust and competitive.

Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

“…We have also extended our results to non-tridiagonal QCQPs by improving a computing method proposed in [29] for simultaneous tridiagonalization. The exactness of the SDP relaxation of the GTRS can be proved by the simultaneous tridiagonalization and our results in section 3.…”

“…Simultaneous tridiagonalization of multiple matrices is an extension of simultaneous diagonalization, and it can be achieved by finding a nonsingular matrix that transforms all matrices to tridiagonal matrices. Recently, Sidje [29] (See also Garvey et al [10]) introduced conditions under which two matrices are simultaneous tridiagonalizable.…”

“…In the beginning of the Sidje's recursive procedure [29], the matrices are initialized as K n := K, M n := M ∈ S n . Then, an appropriate nonsingular matrix 1) .…”

“…To apply the results on tridiagonal QCQPs, the GTRS should be transformed to the tridiagonal QCQP. We improve the simultaneous tridiagonalization technique proposed in [29] so that all tridiagonal QCQPs constructed from the GTRS always satisfy the conditions for the exact SDP relaxation. Similarly, the exactness of other classes of QCQPs can be analyzed by the method presented in this paper.…”

A generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system

“…First, in general, the condition for positivity (4.6) is not sufficient for applications; the condition does not provide concrete ways to choose the parameters for general cases. Second, for applying the proposed algorithm to general (not tridiagonal) matrix pencils, a preconditioning called simultaneous tridiagonalization (see, e.g., [29,30]) is required. In addition to the improvements, comparisons with traditional methods should be discussed.…”