On the structure and geometry of the product singular value decomposition.

*(English)*Zbl 0754.15005The 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.

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.

Reviewer: E.Kreyszig (Ottawa)

##### MSC:

15A18 | Eigenvalues, singular values, and eigenvectors |

##### Keywords:

product singular value decomposition; eigenvalue decompositions; contragredient transformation
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\textit{B. L. R. De Moor}, Linear Algebra Appl. 168, 95--136 (1992; Zbl 0754.15005)

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##### References:

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