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A Note on Skew-Symmetric Matrices

Published online by Cambridge University Press:  03 November 2016

M.P. Drazin
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Extract

Given any skew-symmetric n x n matrix A, we have

det (A - λI)= det (A - λI)′ = det (- A - λI) = (-1)n det (A + λI),

whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A2 is precisely the set of the squares of the eigenvalues of A, it follows that every non-zero eigenvalue of A2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det (A2 - λI) of A2, regarded as a polynomial in λ, is a perfect square if n is even, while, if n is odd, then we may write ϕ(λ) =λ{f(λ)}2 for a suitable polynomial f(λ).

Type
Research Article
Information
The Mathematical Gazette , Volume 36 , Issue 318 , December 1952 , pp. 253 - 255
Copyright
Copyright © The Mathematical Association 1952

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References

page no 253 note * If A is real then these are, in fact, conjugate pure imaginary quantities; however, this point is of no special interest in the present discussion.

page no 253 note † See, e.g., Aitken, A.C., Determinants and Matrices (Edinburgh, 1942), 49,Google Scholar or Ferrar, W. L. , Algebra (Oxford, 1941), 60.Google Scholar

page no 254 note * I have found, during the proof stage ofthis note, that this theorem is, essentially, contained in some work of Jacobson, N., Bull. American Math. Soc., vol. 45 (1939), pp. 745–748.CrossRefGoogle Scholar He showed that, if R, C are given n x n matrices such that RC =C’R, and if R is skew-symmetric and non-singular, then the characteristic function of C is of the form {g(λ)2, here g (A) is a suitable polynomial (and, indeed, g (C)= 0) ; Theorem 1 follows (for non-singular B) on taking R =B, C = AB. However, my proof immediately extends so as to apply to this more general case, and thus furnishes a simple alternative derivation of Jacobson's result.

page no 255 note * In fact, taking A =0, we see at once that the upper sign must always hold; however, we do not need to use this.