invertible matrix , for any zero matrix , and for any block-diagonal matrix. Then defineIt can be shown that as defined here is a generalized inverse of ; in particular a -inverse of . Since it does not in general satisfy , this is not the Moore–Penrose inverse; otherwise we could derive for any choice of matrices, which only holds for certain class of matrices. Suppose , where and . This generalized inverse has the following properties: Quotient SVD A generalized singular ratio of and is . By the above properties, . Note that is diagonal, and that, ignoring the leading zeros, contains the singular ratios in decreasing order. If is invertible, then has no leading zeros, and the generalized singular ratios are the singular values, and and are the matrices of singular vectors, of the matrix . In fact, computing the SVD of is one of the motivations for the GSVD, as "forming and finding its SVD can lead to unnecessary and large numerical errors when is ill-conditioned for solution of equations". Hence the sometimes used name "quotient SVD", although this is not the only reason for using GSVD. If is not invertible, then is still the SVD of if we relax the requirement of having the singular values in decreasing order. Alternatively, a decreasing order SVD can be found by moving the leading zeros to the back: , where and are appropriate permutation matrices. Since rank equals the number of non-zero singular values, . Construction Let be the SVD of , where is unitary, and and are as described, , where and , , where and , by the SVD of , where , and are as described, by a decomposition similar to a QR-decomposition, where and are as described. ThenWe also haveThereforeSince has orthonormal columns, . ThereforeWe also have for each such that thatTherefore , and Applications The GSVD, formulated as a comparative spectral decomposition, has been successfully applied to signal processing and data science, e.g., in genomic signal processing. These applications inspired several additional comparative spectral decompositions, i.e., the higher-order GSVD (HO GSVD) and the tensor GSVD. Second version: weighted single-matrix decomposition The weighted version of the generalized singular value decomposition (GSVD) is a constrained matrix decomposition with constraints imposed on the left and right singular vectors of the singular value decomposition. This form of the GSVD is an extension of the SVD as such. Given