6610-2022-01

(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. 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 \[ \sigma _{\max }(A) = \max _{y \in \mathbb {C}^m,\ x \in \mathbb {C}^n} \frac {|y^*Ax|}{\|x\|_2\|y\|_2}. \]

(Properties of Projectors). Prove algebraically:

1. Show that if \(P \in \mathbb {C}^{m \times m}\) is a nonzero projector, then \(\|P\|_2 \ge 1\).
2. Show that if \(P\) is an orthogonal projector, then \(I - 2P\) is unitary.

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