R: Spectral Decomposition of a Matrix

eigen {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 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 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)

[Package Contents]