The Restricted Singular Value Decomposition: Properties and Applications
Abstract
The restricted singular value decomposition (RSVD) is the factorization of a given matrix, relative to two other given matrices. It can be interpreted as the ordinary singular value decomposition with different inner products in row and column spaces. Its properties and structure, as well as its connection to generalized eigenvalue problems, canonical correlation analysis, and other generalizations of the singular value decomposition, are investigated in detail.
Applications that are discussed include the analysis of the extended shorted operator, unitarily invariant norm minimization with rank constraints, rank minimization in matrix balls, the analysis and solution of linear matrix equations, rank minimization of a partitioned matrix, and the connection with generalized Schur complements, constrained linear and total linear least squares problems with mixed exact and noisy data, including a generalized Gauss–Markov estimation scheme.
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References
1.
L. Autonne, Sur les groupes linéaires, réels et orthogonaux, Bull. Sci. Math., France, 30 (1902), 121–133
2.
E. Beltrami, G. Battagline, E. Fergola, Sulle funzioni bilineari, Giornale di Mathematiche, Vol. 11, 1873, 98–106
3.
ke Björck, Gene H. Golub, Numerical methods for computing angles between linear subspaces, Math. Comp., 27 (1973), 579–594
4.
David Carlson, What are Schur complements, anyway?, Linear Algebra Appl., 74 (1986), 257–275
5.
John S. Chipman, Estimation and aggregation in econometrics: an application of the theory of generalized inversesGeneralized inverses and applications (Proc. Sem., Mat. Res. Center, Univ. Wisconsin, Madison, Wis., 1973), Academic Press, New York, 1976, 549–769. Publ. Math. Res. Center Univ. Wisconsin, No. 32
6.
Chandler Davis, W. M. Kahan, H. F. Weinberger, Norm-preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal., 19 (1982), 445–469
7.
James Weldon Demmel, The smallest perturbation of a submatrix which lowers the rank and constrained total least squares problems, SIAM J. Numer. Anal., 24 (1987), 199–206
8.
B. De Moor, G. H. Golub, Generalized singular value decompositions: A proposal for a standardized nomenclature, Numerical Analysis Project Manuscript, NA-89-03, Department of Computer Science, Stanford University, Stanford, CA, 1989, April, also in ESAT-SISTA Report 1989-10, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, April 1989.
9.
B. De Moor, G. H. Golub, The restricted singular value decomposition: Properties and applications, Numerical Analysis Project Manuscript, NA-89-04, Department of Computer Science, Stanford University, Stanford, CA, 1989, April; also in ESAT-SISTA Report 1989-09, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, April 1989
10.
B. De Moor, On the structure and geometry of the product singular value decomposition, Numerical Analysis Project Manuscript, NA-89-05, Department of Computer Science, Stanford University, Stanford, CA, 1989, May, also in ESAT-SISTA Report 1989-12, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, May 1989
11.
Bart De Moor, Hong Yuan Zha, A tree of generalizations of the ordinary singular value decomposition, Linear Algebra Appl., 147 (1991), 469–500
12.
C. Eckart, G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936), 211–218
13.
L. M. Ewerbring, F. T. Luk, Canonical correlations and generalized SVD; Applications and new algorithms, Proc. SPIE, Vol. 977, Real Time Signal Processing XI, paper 23, 1988
14.
K. V. Fernando, S. J. Hammarling, B. N. Datta, C. R. Johnson, M. A. Kaashoek, R. Plemmons, E. Sontag, A product induced singular value decomposition for two matrices and balanced realisationLinear Algebra in Signal Systems and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988, 128–140
15.
G. H. Golub, C. Van Loan, Matrix computations, Johns Hopkins Series in the Mathematical Sciences, Vol. 3, Johns Hopkins University Press, Baltimore, MD, 1983xvi+476
16.
G. H. Golub, C. Van Loan, An analysis of the total least squares problem, SIAM J. Numer. Anal., 17 (1980), 883–893
17.
G. H. Golub, Alan Hoffman, G. W. Stewart, A generalization of the Eckart-Young-Mirsky matrix approximation theorem, Linear Algebra Appl., 88/89 (1987), 317–327
18.
W. E. Larimore, Identification of nonlinear systems using canonical variate analysis, Proc. 26th Conference on Decision and Control, Los Angeles, CA, 1987, December
19.
C. Lawson, R. Hanson, Solving least squares problems, Prentice-Hall Inc., Englewood Cliffs, N.J., 1974xii+340
20.
Sujit Kumar Mitra, Madan L. Puri, Shorted matrices—an extended concept and some applications, Linear Algebra Appl., 42 (1982), 57–79
21.
M. Z. Nashed, Generalized inverses and applications, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976xiv+1054
22.
C. C. Paige, M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981), 398–405
23.
C. C. Paige, The general linear model and the generalized singular value decomposition, Linear Algebra Appl., 70 (1985), 269–284
24.
R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406–413
25.
R. Penrose, On best approximation solutions of linear matrix equations, Proc. Cambridge Philos. Soc., 52 (1956), 17–19
26.
J. J. Sylvester, Sur la réduction biorthogonale d'une forme linéo-linéaire à sa forme canonique, Comptes Rendus, CVIII (1889), 651–653
27.
J. Vandewalle, B. De Moor, E. Deprettere, A variety of applications of the singular value decompositionSVD and Signal Processing: Algorithms, Applications and Architectures, North-Holland, Amsterdam, 1988, 43–91
28.
Sabine Van Huffel, Joos Vandewalle, Analysis and properties of the generalized total least squares problem $AX\approx B$ when some or all columns in A are subject to error, SIAM J. Matrix Anal. Appl., 10 (1989), 294–315
29.
S. Van Huffel, H. Zha, Restricted total least squares: A unified approach for solving (generalized) (total) least squares problems with(out) equality constraints, ESAT-SISTA Report, 1989-05, Department of Electrical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1989, March
30.
Charles F. Van Loan, Generalizing the singular value decomposition, SIAM J. Numer. Anal., 13 (1976), 76–83
31.
G. A. Watson, The smallest perturbation of a submatrix that lowers the rank of the matrix, IMA J. Numer. Anal., 8 (1988), 295–303
32.
H. Zha, Restricted SVD for matrix triplets and rank determination of matrices, Scientific Report, 89-2, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, Germany, 1989
33.
H. Zha, A numerical algorithm for computing the RSVD for matrix triplets, Scientific Report, 89-1, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, Germany, 1989
Information & Authors
Information
Published In
SIAM Journal on Matrix Analysis and Applications
Pages: 401 - 425
DOI: 10.1137/0612029
ISSN (online): 1095-7162
Copyright
Copyright © 1991 Society for Industrial and Applied Mathematics.
History
Submitted: 8 June 1989
Accepted: 12 September 1990
Published online: 17 July 2006
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