% File src/library/base/man/eigen.Rd
% Part of the R package, http://www.R-project.org
% Copyright 1995-2013 R Core Team
% Distributed under GPL 2 or later

\name{eigen}
\alias{eigen}
\concept{eigen vector}
\concept{eigen value}
\title{Spectral Decomposition of a Matrix}
\description{
  Computes eigenvalues and eigenvectors of numeric (double, integer,
  logical) or complex matrices.
}
\usage{
eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)
}
\arguments{
  \item{x}{a numeric or complex matrix whose spectral decomposition is to
    be computed.  Logical matrices are coerced to numeric.}
  \item{symmetric}{if \code{TRUE}, the matrix is assumed to be symmetric
    (or Hermitian if complex) and only its lower triangle (diagonal
    included) is used.  If \code{symmetric} is not specified, the matrix
    is inspected for symmetry.}
  \item{only.values}{if \code{TRUE}, only the eigenvalues are computed
    and returned, otherwise both eigenvalues and eigenvectors are
    returned.}
  \item{EISPACK}{logical. Defunct and ignored.}
}
\details{
  If \code{symmetric} is unspecified, the code attempts to
  determine if the matrix is symmetric up to plausible numerical
  inaccuracies.  It is faster and surer to set the value yourself.

  Computing the eigenvectors is the slow part for large matrices.

  Computing the eigendecomposition of a matrix is subject to errors on a
  real-world computer: the definitive analysis is Wilkinson (1965).  All
  you can hope for is a solution to a problem suitably close to
  \code{x}.  So even though a real asymmetric \code{x} may have an
  algebraic solution with repeated real eigenvalues, the computed
  solution may be of a similar matrix with complex conjugate pairs of
  eigenvalues.
}
\value{
  The spectral decomposition of \code{x} is returned as components of a
  list with components

  \item{values}{a vector containing the \eqn{p} eigenvalues of \code{x},
    sorted in \emph{decreasing} order, according to \code{Mod(values)}
    in the asymmetric case when they might be complex (even for real
    matrices).  For real asymmetric matrices the vector will be
    complex only if complex conjugate pairs of eigenvalues are detected.
  }
  \item{vectors}{either a \eqn{p\times p}{p * p} matrix whose columns
    contain the eigenvectors of \code{x}, or \code{NULL} if
    \code{only.values} is \code{TRUE}.  The vectors are normalized to
    unit length.

    Recall that the eigenvectors are only defined up to a constant: even
    when the length is specified they are still only defined up to a
    scalar of modulus one (the sign for real matrices).
  }

  If \code{r <- eigen(A)}, and \code{V <- r$vectors; lam <- r$values},
  then \deqn{A = V \Lambda V^{-1}}{A = V Lmbd V^(-1)} (up to numerical
  fuzz), where \eqn{\Lambda =}{Lmbd =}\code{diag(lam)}.
}
\source{
  By default \code{eigen} uses the LAPACK routines \code{DSYEVR},
  \code{DGEEV}, \code{ZHEEV} and \code{ZGEEV} whereas

  LAPACK is from \url{http://www.netlib.org/lapack} and its guide is listed
  in the references.
}
\references{
  Anderson. E. and ten others (1999)
  \emph{LAPACK Users' Guide}.  Third Edition.  SIAM.\cr
  Available on-line at
  \url{http://www.netlib.org/lapack/lug/lapack_lug.html}.

  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
  \emph{The New S Language}.
  Wadsworth & Brooks/Cole.
  Springer-Verlag Lecture Notes in Computer Science \bold{6}.

  Wilkinson, J. H. (1965) \emph{The Algebraic Eigenvalue Problem.}
  Clarendon Press, Oxford.
}

\seealso{
  \code{\link{svd}}, a generalization of \code{eigen}; \code{\link{qr}}, and
  \code{\link{chol}} for related decompositions.

  To compute the determinant of a matrix, the \code{\link{qr}}
  decomposition is much more efficient: \code{\link{det}}.
}
\examples{
eigen(cbind(c(1,-1), c(-1,1)))
eigen(cbind(c(1,-1), c(-1,1)), symmetric = FALSE)
# same (different algorithm).

eigen(cbind(1, c(1,-1)), only.values = TRUE)
eigen(cbind(-1, 2:1)) # complex values
eigen(print(cbind(c(0, 1i), c(-1i, 0)))) # Hermite ==> real Eigenvalues
## 3 x 3:
eigen(cbind( 1, 3:1, 1:3))
eigen(cbind(-1, c(1:2,0), 0:2)) # complex values

}
\keyword{algebra}
\keyword{array}