eigen                  package:base                  R Documentation

Spectral Decomposition of a Matrix

Description:

     Function `eigen' computes eigenvalues and eigenvectors by
     providing an interface to the EISPACK routines `RS', `RG', `CH'
     and `CG'.

     Function `La.eigen' uses the LAPACK routines DSYEV/DSYEVR, DGEEV,
     ZHEEV and ZGEEV.

Usage:

     eigen(x, symmetric, only.values = FALSE)
     La.eigen(x, symmetric, only.values = FALSE,
              method = c("dsyevr", "dsyev"))

Arguments:

       x: a matrix whose spectral decomposition is to be computed.

symmetric: if `TRUE', the matrix is assumed to be symmetric (or
          Hermitian if complex) and only its lower triangle is used. If
          `symmetric' 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.

  method: The LAPACK routine to use in the real symmetric case.

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.

     `La.eigen' is preferred to `eigen' 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.)

     The LAPACK routine DSYEVR is usually substantially faster than
     DSYEV: see <URL:
     http://www.cs.berkeley.edu/~demmel/DOE2000/Report0100.html>. Most
     benefits are seen with an optimized BLAS system.

     Computing the eigenvectors is the slow part for large matrices.

     Using `method="dsyevr"' requires IEEE 754 arithmetic.  Should this
     not be supported on your platform, `method="dsyev"' is used, with
     a warning.

Value:

     The spectral decomposition of `x' is returned as components of a
     list. 

  values: a vector containing the p eigenvalues of `x', sorted in
          decreasing order, according to `Mod(values)' if they are
          complex. 

 vectors: a p * p matrix whose columns contain the eigenvectors of `x',
          or `NULL' if `only.values' is `TRUE'.

          For `eigen(, symmetric = FALSE)' the choice of length of the
          eigenvectors is not defined by LINPACK. 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). 

References:

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

     Anderson. E. and ten others (1999) LAPACK Users' Guide. Third
     Edition. SIAM.
     Available on-line at <URL:
     http://www.netlib.org/lapack/lug/lapack_lug.html>.

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'.

     `capabilities' to test for IEEE 754 arithmetic.

Examples:

     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 <- .Machine$double.eps
     set.seed(123, kind="default")  # force a particular seed
     m <- matrix(round(rnorm(25),3), 5,5)
     sm <- m + t(m) #- symmetric matrix
     em <- eigen(sm); V <- em$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(lam - lam2[i]) < 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)

