A Note on Skew-Symmetric Matrices

Given any skew-symmetric n x n matrix A , we have det ( A - λ I )= det ( A - λ I )′ = det (- A - λ I ) = (-1) n det ( A + λ I ), whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A 2 is precisely the set of the squares of the eigenvalues of A , it follows that every non-zero eigenvalue of A 2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det ( A 2 - λI) of A 2 , 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).