\Header{eigen}{Spectral Decomposition of a Matrix}
\keyword{algebra}{eigen}
\keyword{array}{eigen}
\begin{Description}\relax
This function computes eigenvalues and eigenvectors by providing an
interface to the EISPACK routines \code{RS}, \code{RG}, \code{CH}
and \code{CG}.\end{Description}
\begin{Usage}
\begin{verbatim}
eigen(x, symmetric, only.values=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 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.
\end{ldescription}
\end{Arguments}
\begin{Value}
The spectral decomposition of \code{x} is returned
as components of a list.
\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)}
if they are complex. Complex conjugate pairs have the eigenvalue
with the positive imaginary part first.

\item[\code{vectors}] 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}.  When \code{symmetric = TRUE} the vectors are
normalized to unit length.  Otherwise the choice of length of the
eigenvectors is not defined in the underlying Fortran code.
\end{ldescription}
\end{Value}
\begin{References}\relax
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.\end{References}
\begin{SeeAlso}\relax
\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}}.\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

Meps <- .Alias(.Machine$double.eps)
m <- matrix(round(rnorm(25),3), 5,5)
sm <- m + t(m) #- symmetric matrix
em <- eigen(sm); V <- em$vect
print(lam <- em$values) # ordered DEcreasingly

stopifnot(
 abs(sm %*% V - V %*% diag(lam))          < 60*Meps,
 abs(sm       - V %*% diag(lam) %*% t(V)) < 60*Meps)

##------- Symmetric = FALSE:  -- different to above : ---

em <- eigen(sm, symmetric = FALSE); V2 <- em$vect
print(lam2 <- em$values) # ordered decreasingly in ABSolute value !
                         # and V2 is not normalized (where V is):
print(i <- rev(order(lam2)))
stopifnot(abs(1 - lam2[i] / lam) < 60 * Meps)

zapsmall(Diag <- t(V2) %*% V2) # orthogonal, but not normalized
print(norm2V <- apply(V2 * V2, 2, sum))
stopifnot( abs(1- norm2V / diag(Diag)) < 60*Meps)

V2n <- sweep(V2,2, STATS= sqrt(norm2V), FUN="/")## V2n are now Normalized EV
apply(V2n * V2n, 2, sum)
##[1] 1 1 1 1 1

## Both are now TRUE:
stopifnot(abs(sm %*% V2n - V2n %*% diag(lam2))            < 60*Meps,
          abs(sm         - V2n %*% diag(lam2) %*% t(V2n)) < 60*Meps)

## Re-ordered as with symmetric:
sV <- V2n[,i]
slam <- lam2[i]
all(abs(sm %*% sV -  sV %*% diag(slam))             < 60*Meps)
all(abs(sm        -  sV %*% diag(slam) %*% t(sV)) < 60*Meps)
## sV  *is* now equal to V  -- up to sign (+-) and rounding errors
all(abs(c(1 - abs(sV / V)))       <     1000*Meps) # TRUE (P ~ 0.95)
\end{ExampleCode}
\end{Examples}

