eigen {base}R Documentation

Spectral Decomposition of a Matrix

Description

This function computes eigenvalues and eigenvectors by providing an interface to the EISPACK routines RS, RG, CH and CG.

Usage

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.

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. Complex conjugate pairs have the eigenvalue with the positive imaginary part first.
vectors a p * p matrix whose columns contain the eigenvectors of x, or NULL if only.values is TRUE. When symmetric = TRUE the vectors are normalized to unit length. Otherwise the choice of length of the eigenvectors is not defined in the underlying Fortran code.

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.

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

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(1 - lam2[i] / lam) < 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)