eigen
Spectral Decomposition of a Matrix
Computes eigenvalues and eigenvectors of numeric (double, integer, logical) or complex matrices.
Usage
eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)
Arguments
 x
 a numeric or complex matrix whose spectral decomposition is to be computed. Logical matrices are coerced to numeric.
 symmetric
 if
TRUE
, the matrix is assumed to be symmetric (or Hermitian if complex) and only its lower triangle (diagonal included) is used. Ifsymmetric
is not specified, the matrix is inspected for symmetry.  only.values
 if
TRUE
, only the eigenvalues are computed and returned, otherwise both eigenvalues and eigenvectors are returned.  EISPACK
 logical. Defunct and ignored.
Details
If 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
realworld computer: the definitive analysis is Wilkinson (1965). All
you can hope for is a solution to a problem suitably close to
x
. So even though a real asymmetric 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.
Unsuccessful results from the underlying LAPACK code will result in an
error giving a positive error code (most often 1
): these can
only be interpreted by detailed study of the FORTRAN code.
Value

The spectral decomposition of x is returned as components of a
list with components
 values
 a vector containing the $p$ eigenvalues of
x
, sorted in decreasing order, according toMod(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.  vectors
 either a $p * p$ matrix whose columns
contain the eigenvectors of
x
, orNULL
ifonly.values
isTRUE
. 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 r < eigen(A), and V < r$vectors; lam < r$values,
then A = V \Lambda V^{1}A = V Lmbd V^(1) (up to numerical
fuzz), where \Lambda =Lmbd =diag(lam).
Source
eigen
uses the LAPACK routines DSYEVR
, DGEEV
,
ZHEEV
and ZGEEV
. LAPACK is from http://www.netlib.org/lapack and its guide is listed
in the references.
References
Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM. Available online at http://www.netlib.org/lapack/lug/lapack_lug.html.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. SpringerVerlag Lecture Notes in Computer Science 6.
Wilkinson, J. H. (1965) The Algebraic Eigenvalue Problem. Clarendon Press, Oxford.
See Also
svd
, a generalization of eigen
; qr
, and
chol
for related decompositions.
To compute the determinant of a matrix, the qr
decomposition is much more efficient: det
.
Examples
library(base)
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