Structured Factorizations in Scalar Product Spaces

Mackey, D.S. and Mackey, N. and Tisseur, F. (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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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 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Ms Lucy van Russelt
Date Deposited: 04 Dec 2007
Last Modified: 20 Oct 2017 14:12

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