Spectral Decomposition of a Matrix
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.
|
Description
This function computes eigenvalues and eigenvectors by providing an
interface to the EISPACK routines RS, RG, CH and CG.Value
The spectral decomposition of x is returned
as components of a list.
values
|
a vector containing the p eigenvalues of x, sorted
decreasingly, 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, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe,
V. Klema, C. B. Moler (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)