Prove all eigenvalues of $A^*A$ are non-negative Let $A$ be an $m\times n$ matrix. Prove that all eigenvalues of $A^*A$ are non-negative.
Hint: Pick an eigenvector $x$ of $A^*A$ and consider $x^*A^*Ax$.
$A^*Ax=\lambda x\Rightarrow x^*A^*A=\bar{\lambda}x^*\Rightarrow x^*A^*Ax=\bar{\lambda}x^*x\Rightarrow x^*x(\lambda-\bar{\lambda})=0\Rightarrow\lambda=\bar{\lambda}$