Generalized singular value decomposition

In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition. The two versions differ because one version decomposes two (or more) matrices (much like higher order PCA) and the other version uses a set of constraints imposed on the left and right singular vectors. In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition. The two versions differ because one version decomposes two (or more) matrices (much like higher order PCA) and the other version uses a set of constraints imposed on the left and right singular vectors. The generalized singular value decomposition (GSVD) is a matrix decomposition more general than the singular value decomposition. It was introduced by Van Loan in 1976 and later developed by Paige and Saunders. 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 Let F = R {displaystyle mathbb {F} =mathbb {R} } , or F = C {displaystyle mathbb {F} =mathbb {C} } .Given matrices A ∈ F m × n {displaystyle Ain mathbb {F} ^{m imes n}} and B ∈ F p × n {displaystyle Bin mathbb {F} ^{p imes n}} , their GSVD is given by