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. Ifsymmetricis not specified,isSymmetric(x)is used.- 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, isSymmetric(x)
determines if the matrix is symmetric up to plausible numerical
inaccuracies. It is surer and typically much faster 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
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 a list with components
a vector containing the \(p\) eigenvalues of x,
sorted in decreasing order, according to 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.
either a \(p\times p\) matrix whose columns
contain the eigenvectors of x, or NULL if
only.values is 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).
When only.values is not true, as by default, the result is of S3 class "eigen".
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).
References
Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM. Available on-line 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.
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)
# NOT RUN {
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
# }