Handbook of Linear Algebra, 2nd Edition

Chapter 58

Computation of the Singular Value Decomposition

Alan Kaylor Cline

The University of Texas at Austin

Inderjit S. Dhillon

The University of Texas at Austin

58.1 Singular Value Decomposition

Definitions:

Given a complex matrix A having m rows and n columns, if σ is a nonnegative scalar and u and v are nonzero m- and n-vectors, respectively, such that

$\begin{array}{ccc}Av=\sigma u& \text{and}& A^{*}u=\sigma v,\end{array}$

then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.)

If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t