eigen | R Documentation |
Computes eigenvalues and eigenvectors of numeric (double, integer, logical) or
complex madness
matrices.
## S4 method for signature 'madness'
eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)
x |
|
symmetric |
if |
only.values |
if |
EISPACK |
logical. Defunct and ignored. |
The singular value decomposition of the matrix X
is
X = U D V',
where U
and V
are orthogonal, V'
is V
transposed, and D
is a diagonal matrix with the singular
values on the diagonal.
a list with components
a madness
object of a vector containing
the p
eigenvalues of x
, sorted in decreasing order,
according to Mod(value)
in the assymetric case when they might
be complex (even for real matrices). For real asymmetric matrices
the vector will be complex only if complex conjugate pairs of eigenvalues are
detected.
either a p \times p
matrix whose columns contain the
eigenvectors of x
or NULL
if only.values
is
TRUE
. 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).
If r <- eigen(A)
, and V <- r$vectors; lam <- r$values
, then
A = V Lmbd V^{-1}
(up to numerical fuzz), where Lmbd =diag(lam)
.
Steven E. Pav shabbychef@gmail.com
Izenman, Alan Julian. "Reduced-Rank Regression for the Multivariate Linear Model." Journal of Multivariate Analysis 5, pp 248-264 (1975). https://www.sciencedirect.com/science/article/pii/0047259X75900421
Kato, Tosio. "Perturbation Theory for Linear Operators." Springer (1995). https://www.maths.ed.ac.uk/~v1ranick/papers/kato1.pdf
eigen
.
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