# SCM Repository

[matrix] View of /pkg/man/Schur.Rd
 [matrix] / pkg / man / Schur.Rd # View of /pkg/man/Schur.Rd

Thu Mar 31 21:19:35 2005 UTC (14 years, 3 months ago) by bates
File size: 1757 byte(s)
Create "fallback" methods for any kind of numeric, dense matrix
\name{Schur}
\title{Schur Decomposition of a Matrix}
\usage{
Schur(x, vectors, \dots)
}
\alias{Schur}
\alias{Schur,dgeMatrix,logical-method}
\alias{Schur,dgeMatrix,missing-method}
\alias{Schur,ddenseMatrix,logical-method}
\alias{Schur,ddenseMatrix,missing-method}
\description{
Computes the Schur decomposition and eigenvalues of a square matrix.
}
\arguments{
\item{x}{
numeric or complex square Matrix inheriting from class
\code{"Matrix"}. Missing values (NAs) are not allowed.
}
\item{vectors}{logical.  When \code{TRUE} (the default), the Schur
vectors are computed.
}
\item{\dots}{further arguments passed to or from other methods.}
}
\value{
An object of class \code{c("schur.Matrix", "decomp")} whose
attributes include the eigenvalues, the Schur quasi-triangular form
of the matrix, and the Schur vectors (if requested).
}
\details{
Based on the Lapack functions \code{dgeesx}
}
\section{BACKGROUND}{
If \code{A} is a square matrix, then \code{A = Q T t(Q)}, where
\code{Q} is orthogonal, and \code{T} is upper quasi-triangular
(nearly triangular with either 1 by 1 or 2 by 2 blocks on the
diagonal).
The eigenvalues of \code{A} are the same as those of \code{T},
which are easy to compute. The Schur form is used most often for
computing non-symmetric eigenvalue decompositions, and for computing
functions of matrices such as matrix exponentials.
}
\references{
Anderson, E., et al. (1994).
\emph{LAPACK User's Guide,}
}
\examples{
Schur(Hilbert(9))              # Schur factorization (real eigenvalues)
A <- Matrix(rnorm( 9*9, sd = 100), nrow = 9)
schur.A <- Schur(A)
#mod.eig <- Mod(schur.A\$values) # eigenvalue modulus
#schur.A
}
\keyword{algebra}  