eigen
Eigenvalues and Eigenvectors of a Matrix

### Description

Calculates the eigenvalues and eigenvectors of a square matrix. The eigen function is generic.

### Usage

    eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)

### Arguments

x | a square matrix to decompose. The matrix must not contain missing values (NAs) or any special values of IEEE floating point arithmetic. For more information about IEEE special values, see IEEE Standard for Floating-Point Arithmetic (http://standards.ieee.org/findstds/standard/754-2008.html).
symmetric | a logical value. If TRUE, x is considered symmetric, and the algorithm examines only the lower triangle of x. If this argument is not supplied, isSymmetric(x) is used.
only.values | a logical value. If FALSE (the default) or not present, computes both eigenvalues and eigenvectors. If TRUE, eigenvalues are computed but the value for eigenvectors is NULL.
EISPACK | a logical value. If TRUE, EISPACK is called instead of LAPACK. The default is FALSE.

### Details

Different algorithms are used in the symmetric and non-symmetric cases, so the eigenvectors might not change smoothly from a symmetric matrix to a near-by, nearly symmetric matrix.
In the case of a non-symmetric real matrix, the eigenvalues and eigenvectors might be complex.
Eigenvalues occur frequently in statistics (especially in multivariate analysis) as well as in other fields.
The EXAMPLES section shows a function that uses eigen to compute determinants.
Value
returns a list containing the eigenvalues and eigenvectors of x.

values | a list of nrow(x) eigenvalues in descending order of modulus.
vectors | a matrix with the same dimension as x that returns the eigenvectors corresponding to the eigenvalues in values. The jth column is the eigenvector for the jth element of values. If symmetric = TRUE, then each vector has norm 1. If only.values = TRUE, then each vector is NULL. If y <- eigen(x), and if x is symmetric and real, then the following holds to numerical error:
|  | x == y$vectors %*% diag(y$values) %*% t(y$vectors)
If y <- eigen(x), and if x is symmetric and complex, then the following holds to numerical error:
|  | x == y$vectors %*% diag(y$values) %*% t(Conj(y$vectors))
Note Any dimnames included in the input matrix x are not included in the returned vectors matrix.

Background
If A is a square matrix and Ax == gx,
where g is a numeric value and x is a vector,
then g is an eigenvalue of A
and x is an eigenvector of A.
References
Maindonald, J. H. (1984). Statistical Computation. Wiley, New York.
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
Smith, B. T., Boyle, J. M., Dongarra, J. J., et al. (1976). Lecture Notes in Computer Science, Vol. 6, 2nd Edition. Eigensystem Routines - EISPACK Guide. Springer-Verlag, New York.
Thisted, R. A. (1988). Elements of Statistical Computing. Chapman and Hall, New York.
See Also
chol, cov.mve, function, matrix, prod, qr, svd.
Examples

    amat <- matrix(c(19,8,11,2,18,17,15,19,10), nrow = 3)
    eigen(amat)
    eigen(amat, EISPACK = TRUE)
    x<-matrix(1:12, ncol =2)
    pprcom <- eigen(cov(x)) # robust principal components
    # compute determinant using eigenvalues:
    det <- function(x) prod(eigen(x)$values)
    det(amat)

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