Generalized singular value decomposition

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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 = UΣ1[0,R]Q *

and

B = VΣ2[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, Σ1 and Σ2 are m\times r and p\times r matrices, zero except for the leading diagonals which consist of the real numbers αi and βi respectively, satisfying

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

The ratios σi = αi / βi 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.

[edit] 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]
  • MATLAB documentation [2]