Chemistry - 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 Σ 1 [0, R] Q *

and

B = V Σ 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, $Σ 1$ and $Σ 2$ are $m\times r$ and $p\times r$ matrices, zero except for the leading diagonals which consist of the real numbers $α i$ and $β i$ respectively, satisfying

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

The ratios $σ i = α i / β i$ 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 $A B - 1$ .

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]

Categories: Linear algebra