Diagonal scalings for the eigenstructure of arbitrary pencils

TL;DR

This paper introduces diagonal scaling methods for matrix pencils using Sinkhorn-Knopp algorithms to improve eigenvalue computations, addressing the challenge of balancing row and column norms in arbitrary pencils.

Contribution

It develops new diagonal scaling techniques for arbitrary matrix pencils based on Sinkhorn-Knopp algorithms, extending scaling theory to non-square and nonsquare matrices.

Findings

Scaling improves eigenvalue accuracy.

Methods work for both square and nonsquare pencils.

Provides conditions for zero pattern-based scaling.

Abstract

In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that can not be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn-Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider (1989), we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues.