\HeaderA{eigen}{Spectral Decomposition of a Matrix}{eigen}
\keyword{algebra}{eigen}
\keyword{array}{eigen}
\begin{Description}\relax
Computes eigenvalues and eigenvectors of real or complex matrices.
\end{Description}
\begin{Usage}
\begin{verbatim}
eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a matrix whose spectral decomposition is to be computed.
\item[\code{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[\code{only.values}] if \code{TRUE}, only the eigenvalues are computed
and returned, otherwise both eigenvalues and eigenvectors are
returned.
\item[\code{EISPACK}] logical. Should EISPACK be used (for compatibility with
\R{} < 1.7.0)?
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
By default \code{eigen} uses the LAPACK routines DSYEVR,
DGEEV, ZHEEV and ZGEEV whereas \code{eigen(EISPACK=TRUE)} provides an
interface to the EISPACK routines \code{RS}, \code{RG}, \code{CH}
and \code{CG}.

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.

\code{eigen} is preferred to \code{eigen(EISPACK = TRUE)}
for new projects, but its eigenvectors may differ in sign and
(in the asymmetric case) in normalization. (They may also differ
between methods and between platforms.)

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.
\end{Details}
\begin{Value}
The spectral decomposition of \code{x} is returned as components of a
list with components

\begin{ldescription}
\item[\code{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[\code{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}.

For \code{eigen(, symmetric = FALSE, EISPACK =TRUE)} the choice of
length of the eigenvectors is not defined by EISPACK.  In all other
cases 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).

\end{ldescription}
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Smith, B. T, Boyle, J. M., Dongarra, J. J., Garbow, B. S., Ikebe,Y.,
Klema, V., and  Moler, C. B. (1976).
\emph{Matrix Eigensystems Routines -- EISPACK Guide}.
Springer-Verlag Lecture Notes in Computer Science.

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

Wilkinson, J. H. (1965) \emph{The Algebraic Eigenvalue Problem.}
Clarendon Press, Oxford.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{svd}{svd}}, a generalization of \code{eigen}; \code{\LinkA{qr}{qr}}, and
\code{\LinkA{chol}{chol}} for related decompositions.

To compute the determinant of a matrix, the \code{\LinkA{qr}{qr}}
decomposition is much more efficient: \code{\LinkA{det}{det}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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 Eigen values
## 3 x 3:
eigen(cbind( 1,3:1,1:3))
eigen(cbind(-1,c(1:2,0),0:2)) # complex values

\end{ExampleCode}
\end{Examples}

