Gamma-matrices: a new class of simultaneously diagonalizable matrices

TL;DR

This paper introduces Gamma-matrices, a new class of matrices that are simultaneously diagonalizable, aiding in preconditioning Toeplitz systems and improving eigenvalue clustering for better numerical solutions.

Contribution

The paper defines Gamma-matrices, develops algorithms for their fast computation, and demonstrates their effectiveness in approximating Toeplitz matrices for preconditioning.

Findings

Eigenvalues of preconditioned matrices cluster around zero

Gamma-matrices include symmetric and reverse circulant matrices

Algorithms enable fast matrix-vector multiplication

Abstract

In order to precondition Toeplitz systems, we present a new class of simultaneously diagonalizable real matrices, the Gamma-matrices, which include both symmetric circulant matrices and a subclass of the set of all reverse circulant matrices. We define some algorithms for fast computation of the product between a Gamma-matrix and a real vector and between two Gamma-matrices. Moreover, we illustrate a technique of approximating a real symmetric Toeplitz matrix by a Gamma-matrix, and we show that the eigenvalues of the preconditioned matrix are clustered around zero with the exception of at most a finite number of terms.