Exercise 10

Determine whether the following statements are true or false and explain your reasoning.

1. If $A=U\Sigma V^T$ is a singular value decomposition of $A$, then $G=V(\Sigma^T\Sigma)V^T$ is an orthogonal diagonalization of its Gram matrix.

2. If $A=U\Sigma V^T$ is a singular value decomposition of a rank 2 matrix $A$, then $\vvec_1$ and $\vvec_2$ form an orthonormal basis for the column space $\col(A)$.

3. If $A$ is a symmetric matrix, then its set of singular values is the same as its set of eigenvalues.

4. If $A$ is a $10\times7$ matrix and $\sigma_7 = 4$, then the columns of $A$ are linearly independent.

5. The Gram matrix is always orthogonally diagonalizable.