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On the structure and geometry of the product singular value decomposition. (English) Zbl 0754.15005
The paper proves the product singular value decomposition (PSVD) theorem of K. V. Fernando and 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. A. J. Laub, M. T. Heath, C. C. Paige and 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.

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors
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##### References:
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