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 $m\times n$ matrix $A$ and a $p\times n$ matrix $B$ of real or complex numbers the GSVD is

$A=U\Sigma_1 [ 0, R] Q^*$

and

$B=V\Sigma_2 [ 0, R] Q^*$

where $U,V$ and $Q$ are unitary matrices and $R$ is an upper triangular, nonsingular $r\times r$ matrix, and $r \le n$ is the rank of $[A^*,B^*]$ . Also, $\Sigma_1$ and $\Sigma_2$ are $m\times r$ and $p\times r$ matrices, zero except for the leading diagonals which consist of the real numbers $\alpha_i$ and $\beta_i$ respectively, satisfying

$0 \le \alpha_i,\beta_i\le 1$ and $\alpha_i^2 %2B \beta_i^2 =1$ .

The ratios $\sigma_i=\alpha_i/\beta_i$ 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 [1]
GSVD in Genomic Signal Processing [2]