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.
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Appendix: Basic results for skew-symmetry and orthogonality
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:
and equating real and imaginary parts gives:
Premultiplying by \(\mathbf{x}^\prime \hbox { and } \mathbf{y}^\prime \) and adding, gives that
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
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):
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:
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:
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.
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Gower, J.C. Skew symmetry in retrospect. Adv Data Anal Classif 12, 33–41 (2018). https://doi.org/10.1007/s11634-014-0181-7
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DOI: https://doi.org/10.1007/s11634-014-0181-7
Keywords
- Skew-symmetry
- Canonical analysis of skew-symmetric matrices
- Singular value decomposition of skew matrices
- Hedra
- Bimensions
- Triangle diagrams