linear differential equation - periodic solutions

Under what conditions on the matrix A can the linear differential equation dx/dt = Ax where x = (x1,x2....,xn)^T have periodic solutions?

If and only if the matrix A has pure imaginary eigenvalues.

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Originally Posted by HallsofIvy

If and only if the matrix A has pure imaginary eigenvalues.

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Is that the only condition? A 4 mark question thought it would have abit more to it

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Originally Posted by NFS1

Is that the only condition? A 4 mark question thought it would have abit more to it

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You are probably expected to prove this result...

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Originally Posted by Prove It

You are probably expected to prove this result...

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How would i prove that they have pure imaginary eigenvalues? :S