Let A be a non-square matrix, say m*n for m not equal to n, then C=A*A^t and D =A^t*A are both square matrics, but of different sizes. Show that C and D both have the same non-zero eignvalues.
let be an eigenvalue of then for some non-zero vector note that because otherwise we'd have which is impossible.
now which means that is an eigenvalue of so every non-zero eigenvalue of is an eigenvalue of
proving that every non-zero eigenvalue of is an eigenvalue of is identical and is left for you.