Generalized singular value decomposition

In linear algebra the generalized singular value decomposition (GSVD) is a matrix decomposition more general than the singular value decomposition. It is used to study the conditioning and regularization of linear systems with respect to quadratic semi-norms.

Given an $m\; imes\; n$ matrix $A$ and a $p\; imes\; n$ matrix $B$ of real or complex numbers the GSVD is:$A=USigma\_1\; [\; 0,\; R]\; Q^*$and:$B=VSigma\_2\; [\; 0,\; R]\; Q^*$

where $U,V$ and $Q$ are unitary matrices and $R$ is an upper triangular, nonsingular $r\; imes\; r$ matrix, and $r\; le\; n$ is the rank of $[A^*,B^*]$. Also,$Sigma\_1$ and $Sigma\_2$ are $m\; imes\; r$ and $p\; imes\; r$ matrices, zero except for the leading diagonals which consist of the real numbers $alpha\_i$ and respectively, satisfying

: and .

The ratios are analogous to the singular values. In the important special case, where $B$ is square and invertible, they "are" the singular values, and $U$ and $V$ are the matrices of singular vectors of the matrix $AB^\{-1\}$.

References

* Gene Golub, and Charles Van Loan, Matrix Computations, Third Edition, Johns Hopkins University Press, Baltimore, 1996, ISBN 0-8018-5414-8
* Hansen, Per Christian, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation 4. ISBN 0-89871-403-6
* LAPACK manual [http://www.netlib.org/lapack/lug/node36.html]
* MATLAB documentation [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/gsvd.html]