eigen: Spectral Decomposition of a Matrix

In madness: Automatic Differentiation of Multivariate Operations

Spectral Decomposition of a Matrix

Description

Computes eigenvalues and eigenvectors of numeric (double, integer, logical) or complex madness matrices.

Usage

## S4 method for signature 'madness'
eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)

Arguments

x	madness object representing a numeric 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 (diagonal included) is used. If symmetric is not specified, isSymmetric(x) is used.
only.values	if TRUE, only the eigenvalues are computed and returned, otherwise both eigenvalues and eigenvectors are returned.
EISPACK	logical. Defunct and ignored.

Details

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.

Value

a list with components

values

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.

vectors

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).

Author(s)

Steven E. Pav [e-mail]

References

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

See Also

eigen.

madness documentation built on Aug. 21, 2023, 9:07 a.m.