Computes eigenvalues and eigenvectors of numeric (double, integer, logical) or
complex madness matrices.
# S4 method for madness
eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)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).
madness object representing a numeric matrix
whose spectral decomposition is to be computed.
if TRUE, the matrix is assumed to be symmetric
(or Hermitian if complex) and only its lower triangle (diagonal
included) is used. If symmetric is not specified,
isSymmetric(x) is used.
if TRUE, only the eigenvalues are computed
and returned, otherwise both eigenvalues and eigenvectors are
returned.
logical. Defunct and ignored.
Steven E. Pav shabbychef@gmail.com
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.
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