A Tutorial On The Singular Value Decomposition

Year 1996, Volume: 9 Issue: 1, 1 - 12, 30.06.1996

Abdurrahman Karamancıoğlu Can Özdemir

Abstract

An m y, n real matm A can be factored as ITWV , where V and V are

orthonormal, and W is upper left diagonal. Thîs factorization is called Singular Vaîue

Decomposilion (SVD). The matrlces U, W, and V are usefül in characterizing ıhe malrix A. in

this manuscript geometric characterizations are emphasized. Geometric characîerizations are

anaîyzed in terms ofsubspaces, matrix scaling, cmd norms. We also presenî a numerical viewpoint

for SVD m orcfer to keep the maîerial setf-contained. in the last section we îreat a special problem

where action ofîhe matrix A is restncted to a gıven subspace.

Keywords

Singular values, matrix decomposition, matrix

References

• [1] 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,
• [5] V. C. Klema and A. J. Laub, "The singular value decomposition; Its computation and some applications, " 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 gaussian based 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 and signal 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.
• [11] 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's algorithm for computmg singular value dccomposition, " Numerische Mathematik, pp. 279-300, 1988.