A tutorial on the singular value decomposition

Tekil değer ayrıştırmaya genel bakış

An m x n real matrix A can be factored as; $UWV^T$ y where U and V are orthonormal, and W is upper left diagonal. This factorization is c&lled Singular Value Decomposition (SVDJ. The matrices U, W, and V are useful in characterizing the matrix A. In this manuscript geometric characterizations are emphasized. Geometric characterizations are analyzed in terms of subspaces, matrix scaling, and norms. We also present a numerical viewpoint for SVD in order to keep the material self-contained. In the last section we treat a special problem where action of the matrix A is restricted to a given subspace.

PDF

___

[l] F. M. Caliler and C. A. Desoer, Multıvariahle Feedback Systems, Springcr-Verlag,1982.

[2] D, S. Watkins, Fundamenlals ofMalm Compulalions, John Wiley and Sons, 1991.

[3] C.L. La.wsonandR,J.Haııson,Ao;vın^/,eaî(Sîuares7'ro6/ems,Prentice-Hall, 1974.

[4] G, W. Stewart, Inlroduclion loMatrjx Computalions, Academic Press, 1973,

[5j V. C. Klema and A. J. Laub, "The singular value decomposition; Its computation and someapplications," IEEE Trans. Aulomal. Contr., vo\. 25, pp. 164-176, Apr. 1980.

[6] M. G. Safonov, A. J. Laub and G. L. Hartmann, "Feedback properties ofmultivariable systems:The röle and use ofthe retum difîerence matrix," IEEE Trans. Automat. Contr., vol. 26, pp.47-65,Febr. 1981.

[7) N A. Lehtomaki, N. R. Sandell and M. Athans, "Robustness results in linear-quadratic gaussianbased multivariable control design/'/££'£'7/'ö/îA'. Autöma/. Con//'., vol. 26, pp.75-92, Febr. 1981,

[8] J. Vandewalîc and B. D. Moore, "On the use of singular value decomposition in identification andsignal processing," Numericaî Linear Algehra, Digıtaî Ssgnal Processing and Paralîel Algorithms,Edited by G. H. Golub and P. Van Dooren, Springer-Verlag, 1991.

[9] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C,Cambridge Unİv. Press, 1991.

[10] J. G. F. Francis, "The QR transformation I," Compııler Journal, pp. 265-271, 1961.

(l l] J. G. F. Francis, "The QR transformation II," CompulerJournal, pp. 332-345, 1962.

[12] J. H. Witkinson, The algebraic esgenvalue problem, CIarendon Press, Oxford, 1965.

[13] J. P. Chariier, M. Vanbegin, and P. Van Dooren, "On effıcienl implimentation of Kogbetliantz'salgorithm for computmg singular value dccomposition," Numerische Mathematik, pp. 279-300,1988.