In
linear algebraLinear 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 decompositionIn 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 decompositionIn 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
conditioningIn 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
regularizationIn 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
.