Generalized singular value decomposition

# Generalized singular value decomposition

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Encyclopedia
In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

the generalized singular value decomposition (GSVD) is a matrix decomposition
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...

more general than the singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

. It is used to study the conditioning
Condition number
In the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...

and regularization
Regularization (mathematics)
In mathematics and statistics, particularly in the fields of machine learning and inverse problems, regularization involves introducing additional information in order to solve an ill-posed problem or to prevent overfitting...

of linear systems with respect to quadratic semi-norms.

Given an matrix and a matrix of real or complex numbers the GSVD is
and

where and are unitary matrices and is an upper triangular, nonsingular matrix, and is the rank of . Also,
and are and matrices, zero except for the leading diagonals which consist of the real numbers and respectively, satisfying
and .

The ratios are analogous to the singular values. In the important special case, where is square and invertible, they are the singular values, and and are the matrices of singular vectors, of the matrix .