6610-2022-08

(Rank-One Perturbation of the Identity). If \(u\) and \(v\) are \(n\)-vectors, the matrix \(B = I + uv^*\) is known as a rank-one perturbation of the identity. Show that if \(B\) is nonsingular, then its inverse has the form \(B^{-1} = I + \beta uv^*\) for some scalar \(\beta \), and give an expression for \(\beta \). For what \(u\) and \(v\) is \(B\) singular? If it is singular, what is \(\operatorname {null}(B)\)?

(Properties via SVD).

1. Consider \(A \in \mathbb {C}^{m \times n}\). Define what we mean by the singular value decomposition of \(A\).
2. Prove that any matrix in \(\mathbb {C}^{m \times n}\) is the limit of a sequence of matrices of full rank. Use the 2-norm for your proof.

(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, \[ |\det (A)| \le \prod _{j=1}^m \|\mathbf {a}_j\|_2. \]

(Iterative Solvers).

1. State Gauss-Seidel method for the solution of the linear system \(Ax = b\), where \(A \in \mathbb {R}^{m \times m}\).
2. Show that if \(A\) is a strictly diagonally dominant matrix by rows, the Gauss-Seidel method is convergent for any \(x^{(0)}\).