Skew symmetry in retrospect

Abstract

The paper gives a short account of how I became interested in analysing asymmetry in square tables. The early history of the canonical analysis of skew-symmetry and the associated development of its geometrical interpretation are described.

Appendix: Basic results for skew-symmetry and orthogonality

In this appendix, for convenience we gather together some results on skew-symmetric matrices. Some of these are well known but I do not think we ever gave a formal proof of the form of the singular value decomposition of a skew symmetric matrix, other than in a University of Leiden internal report (Gower and Zielman 1992) which was later published in shortened form as Gower and Zielman (1998) without the material given below.

Theorem 1

If N is a real skew-symmetric matrix, then its eigenvalues are imaginary and occur in conjugate pairs $i\sigma$ and $-\,i\sigma$ corresponding to eigenvectors $\mathbf{x }+ i\mathbf{y}$ and $\mathbf{x} - i \mathbf{y}$, respectively where $\mathbf{x}^\prime \mathbf{x = y}^\prime \mathbf{y }$ and $\mathbf{x}^\prime \mathbf{y} = 0$. When the order of N is odd, there is an additional zero eigenvalue

Proof of Theorem 1

If $\uprho + i\sigma$ is an eigenvalue associated with an eigenvector $\mathbf{x} + i\mathbf{y}$ we have:

$$\begin{aligned} \mathbf{N}(\mathbf{x }+ i\mathbf{y}) = (\uprho + { i}\sigma )(\mathbf{x }+ i\mathbf{y}) \end{aligned}$$

and equating real and imaginary parts gives:

$$\begin{aligned} \left. {{\begin{array}{l} {\mathbf{N}x=\rho \mathbf{x}-\sigma \mathbf{y}} \\ {\mathbf{Ny}=\sigma \mathbf{x}+\rho \mathbf{y}} \\ \end{array} }} \right\} \end{aligned}$$

(5)

Premultiplying by $\mathbf{x}^\prime \hbox { and } \mathbf{y}^\prime$ and adding, gives that

$$\begin{aligned} \mathbf{x}^\prime \mathbf{Nx + y}^\prime \mathbf{Ny} = \uprho (\mathbf{x}^\prime \mathbf{x + y}^\prime \mathbf{y}). \end{aligned}$$

But $\mathbf{x}^\prime \mathbf{Nx = y}^\prime \mathbf{Ny} = 0$ and so $\uprho$ = 0, showing that the non-zero eigenvalues are purely imaginary and $\mathbf{x}^\prime \mathbf{y} = 0$. Premultiplying by $\mathbf{y}^\prime$ and $\mathbf{x}^\prime$ shows that $\sigma \mathbf{x}^\prime \mathbf{x} = \sigma \mathbf{y}^\prime \mathbf{y}$.

Setting $\uprho = 0$ in (5) gives

$$\begin{aligned} \left. {{\begin{array}{l} {\mathbf{N}x=-\sigma \mathbf{y}} \\ {\mathbf{Ny}=\sigma \mathbf{x}} \\ \end{array} }} \right\} \end{aligned}$$

(6)

and hence if $i\sigma$ is an eigenvalue of N satisfying $\mathbf{N}(\mathbf{x} + i\mathbf{y}) =i\sigma (\mathbf{x} + i\mathbf{y})$ then the eigenvector equation $\mathbf{N}(\mathbf{x} - i\mathbf{y}) = -i\sigma (\mathbf{x}-i\mathbf{y})$ is also satisfied.

When N is of odd order there is an extra eigenvalue $\upnu$, say, which is not one of a pair. Because the imaginary pairs cancel one another, the sum of all the eigenvectors must be $\upnu$. Hence, $\upnu = trace(\mathbf{N}) = 0$.

Theorem 2

The singular value decomposition of real skew-symmetric matrix N has the form $\mathbf{U}\varvec{\Sigma }\mathbf{JU}^\prime$ where U is orthogonal and J is defined in Sect. 1.

Proof of Theorem 2

Assume that N has a general singular value decomposition $\mathbf{N} = \mathbf{USV}^\prime$. Then U and V are the eigenvectors of the real symmetric positive semi-definite matrices $\mathbf{NN}^\prime = \mathbf{US}^{2}\mathbf{U}^\prime$ and $\mathbf{N}^\prime \mathbf{N} = \hbox {V}\mathbf{S}^{2}\mathbf{V}^\prime$. Because N is skew-symmetric we have that $\mathbf{NN}^\prime = \mathbf{N}^\prime \mathbf{N} = -\mathbf{N}^{2 }$ and hence from (6):

$$\begin{aligned} \mathbf{NN}^\prime \mathbf{x} = \sigma ^{2}\mathbf{x } \quad \hbox { and } \quad \mathbf{N}^\prime \mathbf{Ny} = \sigma ^{2}\mathbf{y}. \end{aligned}$$

This shows that the singular values of N occur in pairs, corresponding to orthogonal vectors x and y which occur as columns in both U and V. Indeed, rearranging (6) gives:

$$\begin{aligned} \mathbf{N}\left( {\mathbf{x},\mathbf{y}} \right) =\sigma \left( {\mathbf{x},\mathbf{y}} \right) \left( {{\begin{array}{l@{\quad }l} 0&{} 1 \\ {-1}&{} 0 \\ \end{array}}} \right) =\sigma \left( {-\mathbf{y},\mathbf{x}} \right) . \end{aligned}$$

This shows that the columns of V corresponding to the singular value $\sigma$ are the same of those of U but in reverse order and a change of sign. When $n$ is of even order all the singular vectors have the relationship $\mathbf{V} = \mathbf{UJ}^\prime$ and when $n$ odd there is a zero singular value and J has to be augmented by a final unit diagonal value. In both cases, J is orthogonal and so is $\mathbf{UJ}^\prime$, as it has to be for a valid singular value decomposition. Thus, finally the SVD of a skew-symmetric matrix is:

$$\begin{aligned} \mathbf{N} = \mathbf{U}\varvec{\Sigma } \mathbf{JU}^\prime \end{aligned}$$

(7)

where $\mathbf{S} = \varvec{\Sigma } = (\sigma _{1}, \sigma _{1}, \sigma _{2}, \sigma _{2},{\ldots },(0))$ and the singular values are assumed to be in non-increasing order and the final “0” is omitted when $n$ is even.