(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that every periodic matrix is similar to a diagonal matrix.

2. Relevant equations

I use ~ to denote A similar to B as A~B

3. The attempt at a solution

Let A be periodic, and let A^m = I.

If A^m = I this implies A^m ~ I.

Claim: if X~Y implies X^n ~ Y^n for n a positive integer.

By induction.

Base case n = 2.

If X~Y then there exists a Z such that X = ZY(Z^-1).

X^2 = X*X = ZY(Z^-1)ZY(Z^-1) = Z*(Y^2)*(Z^-1).

Inductive step Assume true for some n.

X^(n+1) = (X^n) * X = Z(Y^n)(Z^-1)* ZY(Z^-1) = Z (Y^(n+1))(Z^-1) and we are done.

So since A^m ~ I implies A^(m+1) ~ I ^2 implies A^(m+1) ~ I, but since A^(m+1) = A we have A~I and since I is a diagonal matrix we are done.

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# Homework Help: Similar matrices please check my proof.

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