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More precisely. Is there any way one can tell if a matrix is similar to a skew symmetric matrix by looking at its eigenvalues or even better at the coefficients of its characteristic polynomial ? For instance, if a $2 \times 2$ matrix $A$ is similar to a skew symmetric matrix $S$( that is $A=U^{-1}SU$ ) then $$ tr(A)=0, $$ and $$ tr(A^2)=-2 det(A).$$ These two conditions are necessary but not sufficient. Are there any sufficient conditions?

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$A$ is similar to a skew symmetric matrix if and only if $A$ is diagonalizable with purely imaginary eigenvalues.

Here's a different way to put that: $A$ is similar to a skew symmetric matrix if and only if there exist distinct positive numbers $r_1,\dots,r_k$ such that $$ A \left( A^2 + r_1I \right) \left( A^2 + r_2I \right) \cdots \left( A^2 + r_kI \right) = 0 $$ Any set of values $r_1,\dots,r_k$ gives us a sufficient condition, then.

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  • $\begingroup$ Omno , I think that there is a little mistake about the second equivalence. The condition seems (to me) to be: $A\Pi_j(A^2+r_jI_n)=0$ where the $r_j$ are distinct $>0$. Note that $A^2=0$ does not imply that $A$ is similar to a skew-symmetric. $\endgroup$ – loup blanc Jan 25 '16 at 22:12
  • $\begingroup$ @LoupBlanc excellent catch! Fixed it. $\endgroup$ – Omnomnomnom Jan 25 '16 at 22:34

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