gsvd {geigen}

Generalized Singular Value Decomposition

Description

Computes the generalized singular value decomposition of a pair of matrices.

Usage

gsvd(A,B)

Arguments

A

a matrix with $m$ rows and $n$ columns.

B

a matrix with $p$ rows and $n$ columns.

Details

The matrix A is a $m$-by-$n$ matrix and the matrix B is a $p$-by-$n$ matrix. This function decomposes both matrices; if either one is complex than the other matrix is coerced to be complex.

The Generalized Singular Value Decomposition of numeric matrices $A$ and $B$ is given as

$A = U D_1 [0\, R] Q^T$

and

$B = V D_2 [0\, R] Q^T$

where

$U$

an $m\times m$ orthogonal matrix.

$V$

a $p\times p$ orthogonal matrix.

$Q$

an $n\times n$ orthogonal matrix.

$R$

an $r$-by-$r$ upper triangular non singular matrix and the matrix $[0\, R]$ is an $r$-by-$n$ matrix. The quantity $r$ is the rank of the matrix $\left( \begin{array}{c} A \\B \end{array} \right)$ with $r \le n$.

$D_1$,$D_2$

are quasi diagonal matrices and nonnegative and satisfy $D_1^T D_1 + D_2^T D_2 = I$. $D_1$ is an $m$-by-$r$ matrix and $D_2$ is a $p$-by-$r$ matrix.

The Generalized Singular Value Decomposition of complex matrices $A$ and $B$ is given as

$A = U D_1 [0\, R] Q^H$

and

$B = V D_2 [0\, R] Q^H$

where

$U$

an $m\times m$ unitary matrix.

$V$

a $p\times p$ unitary matrix.

$Q$

an $n\times n$ unitary matrix.

$R$

an $r$-by-$r$ upper triangular non singular matrix and the matrix $[0\, R]$ is an $r$-by-$n$ matrix. The quantity $r$ is the rank of the matrix $\left( \begin{array}{c} A \\B \end{array} \right)$ with $r \le n$.

$D_1$,$D_2$

are quasi diagonal matrices and nonnegative and satisfy $D_1^T D_1 + D_2^T D_2 = I$. $D_1$ is an $m$-by-$r$ matrix and $D_2$ is a $p$-by-$r$ matrix.

For details on this decomposition and the structure of the matrices $D_1$ and $D_2$ see http://www.netlib.org/lapack/lug/node36.html.

Value

The return value is a list containing the following components

A

the upper triangular matrix or a part of $R$.

B

lower part of the triangular matrix $R$ if $k+l>m$ (see below).

m

number of rows of $A$.

k

$r{-}l$. The number of rows of the matrix $R$ is $k{+}l$. The first $k$ generalized singular values are infinite.

l

effective rank of the input matrix $B$. The number of finite generalized singular values after the first $k$ infinite ones.

alpha

a numeric vector with length $n$ containing the numerators of the generalized singular values in the first $(k{+}l)$ entries.

beta

a numeric vector with length $n$ containing the denominators of the generalized singular value in the first $(k{+}l)$ entries.

U

the matrix $U$.

V

the matrix $V$.

Q

the matrix $Q$.

For a detailed description of these items see http://www.netlib.org/lapack/lug/node36.html. Auxiliary functions are provided for extraction and manipulation of the various items.

Source

gsvd uses the LAPACK routines DGGSVD3 and ZGGSVD3 from Lapack 3.8.0. LAPACK is from http://www.netlib.org/lapack. The decomposition is fully explained in http://www.netlib.org/lapack/lug/node36.html.

References

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html. See the section Generalized Eigenvalue and Singular Value Problems (http://www.netlib.org/lapack/lug/node33.html) and the section Generalized Singular Value Decomposition (GSVD) (http://www.netlib.org/lapack/lug/node36.html).

See Also

gsvd.aux

Examples

A <- matrix(c(1,2,3,3,2,1,4,5,6,7,8,8), nrow=2, byrow=TRUE)
B <- matrix(1:18,byrow=TRUE, ncol=6)
A
B

z <- gsvd(A,B)
z