Exercise 9

If $A$ is a matrix and $G=A^TA$ its Gram matrix, remember that $$\xvec\cdot(G\xvec) = \xvec\cdot(A^TA\xvec) = (A\xvec)\cdot(A\xvec) = \len{A\xvec}^2.$$

1. For a general matrix $A$, explain why the eigenvalues of $G$ are nonnegative.

2. Given a symmetric matrix $A$ having an eigenvalue $\lambda$, explain why $\lambda^2$ is an eigenvalue of $G$.

3. If $A$ is symmetric, explain why the singular values of $A$ equal the absolute value of its eigenvalues: $\sigma_j = |\lambda_j|$.