Structured tools for structured matrices.

*(English)*Zbl 1027.15013By 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 |