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How to prove the following:

  1. Prove that an eigenvalue of anti-symmetric matrix is either zero or imaginary.
  2. Prove that the eigenvectors of a symmetric tensor are orthogonal.
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As $A^T=-A$, we conclude $(iA)^*=iA$. – Hagen von Eitzen Jan 31 '13 at 13:03
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This answer is partly based on Hagen von Eitzen's comment.

Since $A^T = -A$, $(iA)^* = iA$; that is, $iA$ is self-adjoint, hence has real eigenvalues; it follows that $A$ has imaginary eigenvalues.

To prove the second point, you can generalize the approach for matrices given by Arturo Magidin here.

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