Singular Value Decomposition

An expansion of a Real $M\times N$ Matrix by Orthogonal Outer Products according to

$$A=\sum_{k=1}^K s_k{\bf u}_k{\bf v}_k^{\rm T},$$ (1)

where $s_1\geq s_2 \geq \ldots \geq 0$,

$$K\equiv \min \{M,N\}$$ (2)

and

$${\bf u}_k^{\rm T}{\bf u}_{k'} = {\bf v}_l^{\rm T}{\bf v}_{k'} =\delta_{kk'}.$$ (3)

Here $\delta_{ij}$ is the Kronecker Delta and ${\hbox{\sf A}}^{\rm T}$ is the Matrix Transpose.

See also Cholesky Decomposition, LU Decomposition, QR Decomposition

References

Nash, J. C. ``The Singular-Value Decomposition and Its Use to Solve Least-Squares Problems.'' Ch. 3 in Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 30-48, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Singular Value Decomposition.'' §2.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 51-63, 1992.