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.