Generalized singular value decomposition

In Linear algebra the Generalized singular value decomposition (GSV) is a matrix decomposition more general than the singular value decomposition, used to study the conditioning and for regularization of linear systems with respect to quadratic semi-norms.

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

$A=U\Sigma _{1}[0,R]Q^{*}$

and

$A=V\Sigma _{2}[0,R]Q^{*}$

where

• $U,V$ and $Q$ are unitary
• $R$ is an upper triangular, nonsingular $r\times r$ matrix, where $r\leq n$ is the rank of $[A^{*},B^{*}]$
• $\Sigma _{1}$ 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 $\beta _{i}$ respecively, satisfying $0\leq \alpha _{i},\beta _{i}\leq 1$ and $\alpha _{i}^{2}+\beta _{i}^{2}=1$.

The ratios $\sigma _{i}=\alpha _{i}/\beta _{i}$ are analogous to the singular values, and in an important special case where $B$ is square and invertable then 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: 0801854148
• 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]
• MATLAB documentation [2]