You are here: MIMS > EPrints
MIMS EPrints

2006.44: Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras

2006.44: D. Steven Mackey, Niloufer Mackey and Françoise Tisseur (2006) Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras.

There is a more recent version of this eprint available. Click here to view it.

Full text available as:

PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
282 Kb

Abstract

Given a class of structured matrices $\Sb$, we identify pairs of vectors $x,b$ for which there exists a matrix $A\in\Sb$ such that $Ax=b$, and also characterize the set of all matrices $A\in\Sb$ mapping $x$ to $b$. The structured classes we consider are the Lie and Jordan algebras associated with orthosymmetric scalar products. These include (skew-)symmetric, (skew-)Hamiltonian, pseudo (skew-)Hermitian, persymmetric and perskew-symmetric matrices. Structured mappings with extremal properties are also investigated. In particular, structured mappings of minimal rank are identified and shown to be unique when rank one is achieved. The structured mapping of minimal Frobenius norm is always unique and explicit formulas for it and its norm are obtained. Finally the set of all structured mappings of minimal 2-norm is characterized. Our results generalize and unify existing work, answer a number of open questions, and provide useful tools for structured backward error investigations.

Item Type:MIMS Preprint
Uncontrolled Keywords:Lie algebra, Jordan algebra, scalar product, bilinear form, sesquilinear form, orthosymmetric, adjoint, structured matrix, backward error, Hamiltonian, skew-Hamiltonian, Hermitian, complex symmetric, skew-symmetric, persymmetric, perskew-symmetric, minimal rank, minimal Frobenius norm, minimal 2-norm.
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.44
Deposited By:Dr Françoise Tisseur
Deposited On:21 April 2006

Available Versions of this Item

Download Statistics: last 4 weeks
Repository Staff Only: edit this item