6610-2024-08

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

(Properties via SVD). Suppose \(A \in \mathbb {C}^{m \times n}\) has an SVD, \(A = U\Sigma V^*\), where \(u_j\) are the left singular vectors, \(U = \{u_j\}\), \(v_j\) are the right singular vectors, \(V = \{v_j\}\), and \(\sigma _j\) are the singular values. Assume rank of the matrix \(A\) is \(r\). For any \(\nu \) with \(0 \le \nu \le r\), define \[ A_\nu = \sum _{j=1}^{\nu }\sigma _j u_j v_j^*, \] (if \(\nu = p = \min \{m,n\}\), define \(\sigma _{\nu +1} = 0\)). Show that, \[ \|A - A_\nu \|_2 = \inf _{\substack {B \in \mathbb {C}^{m \times n}\\ \operatorname {rank}(B) \le \nu }} \|A - B\|_2 = \sigma _{\nu +1}. \]

(Eigenvalues and Eigenvectors).

1. State the Rayleigh Quotient Iteration Algorithm to find eigenvalues/eigenvectors for general real symmetric matrix \(A \in \mathbb {R}^{m \times m}\).
2. Prove that the spectrum (the set of all eigenvalues) of any matrix \(A \in \mathbb {C}^{m \times m}\) is contained in the union of the following \(m\) disks, \(D_i\), in the complex plane: \[ D_i = \left \{z \in \mathbb {C} : |z - a_{ii}| \le \sum _{\substack {j=1\\ j\ne i}}^m |a_{ij}|\right \},\qquad 1 \le i \le m. \]

(Properties of the SPD Matrices) Let \(A \in \mathbb {R}^{n \times n}\) be a symmetric positive definite matrix (spd). Show that there exists a unique upper triangular matrix \(H \in \mathbb {R}^{n \times n}\) with positive diagonal entries such that, \(A = H^T H\).