Structured tools for structured matrices. (English) Zbl 1027.15013

Mackey, D. Steven; Mackey, Niloufer; Tisseur, Françoise

Electron. J. Linear Algebra 10, 106-145 (2003).

By structured matrices the authors mean the elements of a fixed real or complex classical group other than the general linear group. They distinguish 8 cases, the relevant classical group being explicitly given in each one. The paper is motivated by the general problem of simplifying a given vector by multiplication with matrices of given structure. The tools provided for the purpose are certain structured matrices of simple form that enable such simplification to be carried out step by step. Among such tools are, for example, structured matrices whose spaces of fixed points have codimension 1 or 2. The overall aim of the paper is to provide a comprehensive library of structured tools available for general use. Besides listing the tools at considerable length, the authors comment in detail on how to use them and where they are likely to find application. Analysis of the numerical behaviour of the proposed methods is deferred to a later paper.

Reviewer: G.E.Wall (Sydney)

MSC:

15B57	Hermitian, skew-Hermitian, and related matrices
15A63	Quadratic and bilinear forms, inner products
65F15	Numerical computation of eigenvalues and eigenvectors of matrices
65F30	Other matrix algorithms (MSC2010)
65F35	Numerical computation of matrix norms, conditioning, scaling

Keywords:

structured matrices; matrix groups; Givens rotations; Householder reflections; complex orthogonal; complex symplectic; conjugate symplectic; sesquilinear form; automorphism groups; Jordan algebra; Lie algebra; real perplectic; complex pseudo-orthogonal; scalar product; bilinear form; pseudo-unitary

Software:

Algorithm 800; LAPACK