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2007.220: Structured Factorizations in Scalar Product Spaces

2007.220: D.S. Mackey, N. Mackey and F. Tisseur (2006) Structured Factorizations in Scalar Product Spaces. SIAM Journal on Matrix Analysis and Applications, 27 (3). pp. 821-850. ISSN 1095-7162

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DOI: 10.1137/040619363

Abstract

Let $A$ belong to an automorphism group, Lie algebra, or Jordan algebra of a scalar product. When $A$ is factored, to what extent do the factors inherit structure from $A$? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general $A$, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors, and singular values that persists across a wide range of scalar products.

A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes are given.

Item Type:Article
Uncontrolled Keywords:automorphism group; adjoint; factorization; symplectic; Hamiltonian; pseudo-orthogonal; polar decomposition; matrix sign function; matrix square root; generalized polar decomposition; eigenvalues; Lie group; eigenvectors; singular values; structure preservation; Lie algebra; Jordan algebra; bilinear form; sesquilinear form; scalar product; indefinite inner product; orthosymmetric
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2007.220
Deposited By:Ms Helen Kirkbright
Deposited On:04 December 2007

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