1. If A=UΣV∗ is a singular value decomposition of a square matrix A, then A is invertible if and only if all diagonal entries of Σ are nonzero. Assuming that A is invertible, write A−1 in terms of factors of the singular value decomposition of A.
2. If all singular values of A∈Fm×n with mn are positive, is A∗A invertible?
