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

Note

To compute the determinant of a matrix (do you really need it?), it is much more efficient to use the QR decomposition, see qr.

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.

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

all(abs(sm %*% V - V %*% diag(lam))          < 60*Meps)
all(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)))
all(abs(1 - lam2[i] / lam) < 60 * Meps)# [1] TRUE

zapsmall(Diag <- t(V2) %*% V2) # orthogonal, but not normalized
print(norm2V <- apply(V2 * V2, 2, sum))
all( abs(1- norm2V / diag(Diag)) < 60*Meps) #> TRUE

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:
all(abs(sm %*% V2n - V2n %*% diag(lam2))            < 60*Meps)
all(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)
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