skewsymmetry {asymmetry}

Decompose an Asymmetric Matrix into Symmetric and Skew-symmetric Components

Description

The decomposition of an asymmetric matrix into a symmetric matrix and a skew-symmetric matrix is an elementary result from mathematics that is the cornerstone of this package. The decomposition into a skew-symmetric and a symmetric component is written as: Q=S+A, where Q is an asymmetric matrix, S is a symmetric matrix, and A is a skew-symmetric matrix. This decomposition provides a justification for separate analyses of S and A. This decomposition is a useful tool for data analysis and graphical representation by areas. A second application is to the study of an asymmetric matrix of residuals, obtained after fitting a MDS model.

Usage

skewsymmetry(x)

Arguments

x

Asymmetric matrix

Value

S

The symmetric part of the matrix

A

The skew-symmetric part of the matrix

linear

The row means of the skew-symmetric matrix, this amounts to fitting a linear model with row and column effects to the skew-symmetric matrix

sv

The singular vectors of the skew-symmetric matrix

sval

a vector containing the singular values of the skew-symmetric part of the data matrix

nobj

The number of objects

See Also

plot.skewsymmetry

Examples

data("Englishtowns")
Q <- skewsymmetry(Englishtowns)
# the skew-symmetric part
Q$A