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.
This is the latest version of this eprint.
Full text available as:
 | PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader 278 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-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 |
Available Versions of this Item
- Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras (deposited 12 June 2007) [Currently Displayed]
Download Statistics: last 4 weeks
Repository Staff Only: edit this item