6610-2024-01

(QR and Cholesky Factorizations). Let \(A\) be a nonsingular square matrix and let \(A = QR\) and \(A^*A = U^*U\) be the QR factorization of \(A\) and the Cholesky factorization of \(A^*A\), respectively. Assume that the usual normalizations \(r_{jj}, u_{jj} > 0\) are in effect. Is it true or false that \(R = U\)? Justify your answer with mathematical details.

(Properties via SVD). Suppose \(A \in \mathbb {C}^{m \times m}\) has an SVD, \(A = U\Sigma V^*\). Find an eigenvalue decomposition of the \(2m \times 2m\) hermitian matrix (using information about SVD for \(A\)), \[ \begin {bmatrix} 0 & A^* \\ A & 0 \end {bmatrix}. \]

(Properties of Projectors). Prove that a projector \(P\) is orthogonal if and only if \(P = P^*\).

(Properties of Gaussian Elimination). Show that for Gaussian elimination with partial pivoting applied to any matrix \(B \in \mathbb {C}^{n \times n}\), the growth factor \(\rho \) satisfies \(\rho \le 2^{n-1}\). Recall that the growth factor is defined as, \[ \rho = \frac {\max _{i,j}|u_{ij}|}{\max _{i,j}|b_{ij}|}, \] where \(u_{i,j}\) are the entries of the upper triangular matrix that results from performing Gaussian elimination on \(B\).