MATRIX NEARNESS PROBLEMS AND APPLICATIONS

A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rank-deficiency and instability.

Abstract

A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rank-deficiency and instability. Theoretical results and computational methods are described. Applications of nearness problems in areas including control theory, numerical analysis and statistics are outlined. Key words. matrix nearness problem, matrix approximation, symmetry, positive definiteness, orthogonality, normality, rank-deficiency, instability. AMS subject classifications. Primary 65F30, 15A57. 1 Introduction Consider the distance function (1:1) d(A) = min \\Phi kEk : A + E 2 S has property P \\Psi ; A 2 S; where S denotes C m\\Thetan or R m\\Thetan , k \\Delta k is a matrix norm on S, and P is a matrix property which defines a subspace or compact sub...