\name{schur}
\title{Schur Decomposition of a Matrix}
\usage{
schur.Matrix(x, vectors=TRUE)
}
\alias{schur}
\alias{schur.Matrix}
\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.
    }
}
\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,}
2nd edition, SIAM, Philadelphia.
}
\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}