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Let $A$ be a real $N\times N$ matrix. If $A^k$ is symmetric for some $k>0$, does that give away something about $A$.

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If you mean in terms of symmetry than $A$ need not be symmetric if $A^k$ is for some $k\ge 2$. –  flavio Dec 11 '12 at 10:03
    
may be in terms of symmetry or may be in terms of its trace or eigenvalues or something. –  dineshdileep Dec 11 '12 at 10:05

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Since $A^k$ is symmetric, it has real eigenvalues. So, at least you know that if $\lambda=re^{i\theta}$ is an eigenvalue of $A$, you must have $\theta=im\pi/k$ for some integer $m$.

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