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Calculating the Singular Values and Pseudo-Inverse of a Matrix

G. Golub and W. Kahan
Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis
Vol. 2, No. 2 (1965), pp. 205-224
Stable URL: http://www.jstor.org/stable/2949777
Page Count: 20
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Calculating the Singular Values and Pseudo-Inverse of a Matrix
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Abstract

A numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form Σ = U* AV, thus exhibiting A's singular values on Σ's diagonal. The scheme first transforms A to a bidiagonal matrix J, then diagonalizes J. The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods. Some applications are mentioned, in particular the use of the pseudo-inverse AI = VΣIU* to solve least squares problems in a way which dampens spurious oscillation and cancellation.

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