An alternative recursive approach to functions of simple triangular matrices

TL;DR

This paper introduces a recursive method for computing functions of simple triangular matrices, avoiding eigenvector calculations and enabling simultaneous computation of multiple functions, with applications in matrix semigroups.

Contribution

It presents a novel recursive approach for matrix functions of triangular matrices with simple spectra, eliminating the need for eigenvector knowledge.

Findings

Recursive method simplifies computation of matrix functions.

Approach applies to multiple functions and semigroups simultaneously.

No eigenvector knowledge required for the method.

Abstract

The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the context of matrix semigroups and their generators. Here, we look at matrix functions of triangular matrices, where a recursive approach is possible when the matrix has simple spectrum. The special feature is that no knowledge of eigenvectors is required, and that the same recursion applies to the computation of multiple functions or semigroups simultaneously.