1. ## Matrix problem, help please

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.

2. Originally Posted by pdnhan

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 $\lambda \neq 0$ be an eigenvalue of $C.$ then $AA^Tv=Cv=\lambda v,$ for some non-zero vector $v.$ note that $w=A^Tv \neq 0,$ because otherwise we'd have $0=Aw=\lambda v,$ which is impossible.

now $Dw=A^TAw=\lambda A^Tv=\lambda w,$ which means that $\lambda$ is an eigenvalue of $D.$ so every non-zero eigenvalue of $C$ is an eigenvalue of $D.$

proving that every non-zero eigenvalue of $D$ is an eigenvalue of $C$ is identical and is left for you.