Qualifying Exam Analysis of Numerical Methods I, January 2025

(Matrix Norms). Show that \[ \|A\|_1 = \max _{j=1,\ldots ,n}\sum _{i=1}^n |a_{ij}|. \]

(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).

1. Consider \(A \in \mathbb {C}^{m \times n}\). Define what we mean by the singular value decomposition of \(A\).
2. Show that the rank of \(A\) is \(r\), the number of nonzero singular values.
3. Show that the largest singular value of \(A\), call it here \(\sigma _{\max }(A)\), satisfies the relation (with \(x\) and \(y\) are being nonzero vectors below), \[ \sigma _{\max }(A) = \max _{y \in \mathbb {C}^m,\ x \in \mathbb {C}^n} \frac {|y^*Ax|}{\|x\|_2\|y\|_2}. \]

(Hadamard’s Inequality). Prove algebraically: Let \(A \in \mathbb {C}^{m \times m}\) and let \(\mathbf {a}_j\) denote the \(j^{\text {th}}\) column of \(A\). Then, show that, \[ |\det (A)| \le \prod _{j=1}^m \|\mathbf {a}_j\|_2. \]