skewsymmetry function

Decompose an Asymmetric Matrix into Symmetric and Skew-symmetric Components

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.

skewsymmetry(x)

Arguments

• x: Asymmetric matrix

Returns

• 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

Examples

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

See Also

plot.skewsymmetry