(Solved): Suppose \( \mathbf{A} \in \mathbb{C}^{m \times n}, \mathbf{B} \in \mathbb{C}^{n \times m} \), and ...

Suppose \( \mathbf{A} \in \mathbb{C}^{m \times n}, \mathbf{B} \in \mathbb{C}^{n \times m} \), and \( m \leq n \). (a) We know \( \mathbf{A B} \) and \( \mathbf{B A} \) are not in general similar. However, we can still say something about their spectrum. Show that the \( n \) eigenvalues of \( \mathbf{B A} \) are the \( m \) eigenavlues of \( \mathbf{A B} \) together with \( n-m \) zeros. (b) Suppose \( \mathbf{A} \in \mathbb{C}^{n \times n} \) is invertible and \( \mathbf{x}, \mathbf{y} \in \mathbb{C}^{n} \). Use the first part of this question and show that \[ \operatorname{det}\left(\mathbf{A}+\mathbf{x} \mathbf{y}^{\top}\right)=\operatorname{det}(\mathbf{A})\left(1+\mathbf{y}^{\top} \mathbf{A}^{-1} \mathbf{x}\right) \]