Exercise 4

Suppose that $A$ is an $m\times n$ matrix and that $B=A^TA$.

1. Explain why $B$ is orthogonally diagonalizable.

2. Explain why $\vvec\cdot(B\vvec) = \len{A\vvec}^2$.

3. Suppose that $\uvec$ is an eigenvector of $B$ with associated eigenvalue $\lambda$ and that $\uvec$ has unit length. Explain why $\lambda = \len{A\uvec}^2$.

4. Explain why the eigenvalues of $B$ are nonnegative.

5. If $C$ is the covariance matrix associated to a demeaned dataset, explain why the eigenvalues of $C$ are nonnegative.