In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD). The two versions differ because one version decomposes two matrices (somewhat like the higher-order or tensor SVD) and the other version uses a set of constraints imposed on the left and right singular vectors of a single-matrix SVD. First version: two-matrix decomposition The generalized singular value decomposition (GSVD) is a matrix decomposition on a pair of matrices which generalizes the singular value decomposition. It was introduced by Van Loan in 1976 and later developed by Paige and Saunders, which is the version described here. In contrast to the SVD, the GSVD decomposes simultaneously a pair of matrices with the same number of columns. The SVD and the GSVD, as well as some other possible generalizations of the SVD, are extensively used in the study of the conditioning and regularization of linear systems with respect to quadratic semi-norms. In the following, let , or . Definition The generalized singular value decomposition of matrices and iswhere is unitary, is unitary, is unitary, is unitary, is real diagonal with positive diagonal, and contains the non-zero singular values of in decreasing order, , is real non-negative block-diagonal, where with , , and , is real non-negative block-diagonal, where with , , and , , , , . We denote , , , and . While is diagonal, is not always diagonal, because of the leading rectangular zero matrix; instead is "bottom-right-diagonal". Variations There are many variations of the GSVD. These variations are related to the fact that it is always possible to multiply from the left by where is an arbitrary unitary matrix. We denote , where is upper-triangular and invertible, and is unitary. Such matrices exist by RQ-decomposition. . Then is invertible. Here are some variations of the GSVD: MATLAB (gsvd): LAPACK (LA_GGSVD): Simplified: Generalized singular values A generalized singular value of and is a pair such that We have By these properties we can show that the generalized singular values are exactly the pairs . We haveTherefore This expression is zero exactly when and for some . In, the generalized singular values are claimed to be those which solve . However, this claim only holds when , since otherwise the determinant is zero for every pair ; this can be seen by substituting above. Generalized inverse Define for any