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, DGEEV, ZHEEV and ZGEEV.

Usage

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

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.

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.

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\times 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 (1995) LAPACK Users' Guide. Second Edition. SIAM.

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 <- .Alias(.Machine$double.eps)
m <- matrix(round(rnorm(25),3), 5,5)
sm <- m + t(m) #- symmetric matrix
em <- eigen(sm); V <- em$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 base version 1.3.1 ]