On the structure and geometry of the product singular value decomposition
type: Item
instance of: scholarly article
title: On the structure and geometry of the product singular value decomposition
zbMATH Open document ID: 0754.15005
DOI: 10.1016/0024-3795(92)90290-Q
published in: Linear Algebra and its Applications
publication date: 1992-09-27T00:00:00Z
review text: The paper proves the product singular value decomposition (PSVD) theorem of \textit{K. V. Fernando} and \textit{S. Hammarling} [Linear algebra in signals, systems, and control, Proc. SIAM Conf., Boston/Mass. 1986, 128- 140 (1988; Zbl 0667.65034)], which states that any pair of real matrices \(A_{m\times n}\) and \(B_{p\times n}\) can be factorized as \(A=U_ AS_ AX^ T\), \(B=U_ BS_ BX^{-1}\), all factors real, \(U_ A\), \(U_ B\) orthogonal, \(X\) square, \(S_ A\), \(S_ B\) of a somewhat more complicated form. The proof exploits the close relation of the PSVD with the ordinary singular value decomposition (OSVD) of \(AB^ TBA^ T\) and the eigenvalue decompositions of \(AA^ TBB^ T\) and \(BB^ TAA^ T\). Here, \(X\) is not unique and the paper gives a detailed characterization of the nonuniqueness properties of the PSVD, in particular those of a so- called contragredient transformation of \(A^ TA\) and \(B^ TB\) [cf. \textit{A. J. Laub}, \textit{M. T. Heath}, \textit{C. C. Paige} and \textit{R. C. Ward}, IEEE Trans. Autom. Control AC-32, 115-122 (1987; Zbl 0624.93025)]. This includes a construction of the PSVD from four OSVDs obtainable from \(A\) and \(B\). The structure of the contragredient transformation is then related to the geometry of subspaces related to \(A\) and \(B\), giving an interpretation in terms of principal angles between subspaces. Improved results (generalized \(QR\), etc.) will be given in a forthcoming paper co-authored by P. Van Dooren.
reviewed by: Erwin O. Kreyszig
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 57565
zbMATH Keywords: product singular value decomposition
zbMATH Keywords: eigenvalue decompositions
zbMATH Keywords: contragredient transformation
author: B. De Moor
MaRDI profile type: Publication
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full work available at URL: 0024-3795(92)90290-q
OpenAlex ID: W2042815084
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description: scientific article; zbMATH DE number 57565