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I am a graduate student working on geometric analysis. Recently I am doing some problem concerning complex skew symmetric matrix. I want some properties about complex skew symmetric matrix which are invariant under similarity transformation. I have found the canonical Jordan form and canonical form of unitary congruent in Horn and Johnson's matrix analysis.

So do there exists canonical form of unitary similarity transformation for complex skew symmetric matrix?

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Note that a unitary similarity transformation $M\mapsto UMU^{-1}$ will not in general preserve the skew-symmetry of $M$. To preserve the skew-symmetry the normal form is defined as $M=UD\bar{U}^{-1}$, with unitary $U$, and then $D$ can be chosen real tridiagonal, $$D=\begin{pmatrix} 0 &\alpha_1\\ -\alpha_1&0\end{pmatrix}\oplus \begin{pmatrix} 0 &\alpha_2\\ -\alpha_2&0\end{pmatrix}\oplus\cdots\oplus \begin{pmatrix} 0 &\alpha_n\\ -\alpha_n&0\end{pmatrix},$$ with real $\alpha_1,\alpha_2,\ldots\alpha_n\geq 0$. [proof]

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  • $\begingroup$ Thanks very much for your answer. Your answer is unitary congruence in Horn and Johnson's book but in fact I don't want to preserve skew-symmetry. I need properties which is invariant under similarity transformation. $\endgroup$
    – hanbf
    Commented Jul 9, 2021 at 14:01
  • $\begingroup$ @hanbf: You want to determine which complex square matrices are similar to skew symmetric ones. Do I understand correctly? Or do you want to arrange some normal form for those complex square matrices which are similar to skew symmetric ones? $\endgroup$
    – Ben McKay
    Commented Jul 10, 2021 at 12:01
  • $\begingroup$ I actually want some normal form for those complex square matrices which are unitarily similar to complex skew symmetric ones. $\endgroup$
    – hanbf
    Commented Jul 10, 2021 at 15:44
  • $\begingroup$ According to the Schur decomposition, everything unitarily equivalent to a skew-symmetric matrix will be of the form D+N, where (a) D is a diagonal matrix with eigenvalues consisting of 0 and pairs of eigenvalues with the same multiplicity that sum to 0 and (b) N is an upper triangular matrix with 0's on the diagonal. Is every matrix that satisfies (a) and (b) unitarily equivalent to a skew-symmetric matrix? $\endgroup$
    – Derek
    Commented Apr 6, 2022 at 10:49

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