eigen {base} | R Documentation |
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.
eigen(x, symmetric, only.values = FALSE) La.eigen(x, symmetric, only.values = FALSE)
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. |
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.
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). |
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.
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
.
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)