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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 (2007) Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras.

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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-1 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:12 June 2007

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