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Polar and Singular Value Decomposition Theorems

In: Thinking in Problems

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  • Alexander A. Roytvarf

Abstract

There exists a very powerful set of techniques for dealing with sets of equations or matrices that are either singular or numerically very close to singular. In many cases where Gaussian elimination and triangle decomposition fail to give satisfactory results, this set of techniques, known as singular value decomposition (SVD) will solve it, in the sense of giving you a useful numerical answer. In this chapter, the reader will see that SVD of any matrix is always realizable. Applications of SVD [and a related tool known as polar decomposition (PD), or canonical factorization, generalizing a complex number’s factorization by its modulus and phase factor] are not restricted to numerical analysis; other application examples are presented in this chapter. More advanced readers who wish to familiarize themselves with infinite-dimensional versions of SVD and PD and their various applications may consult the guide to the literature that is provided.

Suggested Citation

  • Alexander A. Roytvarf, 2013. "Polar and Singular Value Decomposition Theorems," Springer Books, in: Thinking in Problems, edition 127, pages 55-84, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8406-8_6
    DOI: 10.1007/978-0-8176-8406-8_6
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