Abstract
Abstract Generalized matrix functions were first introduced in [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra, 1(2), 1973, pp. 163-171]. Recently, it has been recognized that these matrix functions arise in a number of applications, and various numerical methods have been proposed for their computation. The exploitation of structural properties, when present, can lead to more efficient and accurate algorithms. The main goal of this paper is to identify structural properties of matrices which are preserved by generalized matrix functions. In cases where a given property is not preserved in general, we provide conditions on the underlying scalar function under which the property of interest will be preserved by the corresponding generalized matrix function.
Highlights
A generalized matrix function (GMF) is a type of matrix function that is de ned in terms of the singular value decomposition (SVD), and that can be applied to rectangular matrices
Some of the properties preserved by standard matrix functions are not preserved by generalized matrix functions; there exist structural properties preserved by generalized matrix functions which are not preserved by standard matrix functions
In this paper we have identi ed several types of matrix properties that are preserved under generalized matrix functions
Summary
A generalized matrix function (GMF) is a type of matrix function that is de ned in terms of the singular value decomposition (SVD), and that can be applied to rectangular matrices. GMFs arise naturally in the context of Hamiltonian dynamical systems [7, 8], such as the wave equation on graphs [5]. This realization has prompted several authors to take a new look at GMFs, leading to several papers, both theoretical and computational in nature [1, 3, 5, 15]. In the case of standard matrix functions, advance knowledge of the structural properties of f (A) can lead to more accurate and e cient algorithms; for example, when A is a triangular Toeplitz matrix and f is a function such that f (A) is de ned, the fact that f (A) is triangular and Toeplitz can lead to signi cant savings when computing it. It is necessary to systematically investigate the structural properties that are preserved by GMFs and, more generally, the interplay between matrix structures and GMFs
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