91Ó°ÊÓ

Show by a suitable example that in general \((A B)^{+} \neq B^{+} A^{+} .\)

Prove that if \(A\) is normal, then \(A\) and \(A^{*}\) have the same eigenvectors.

$$ \begin{array}{l} \text { What is the characteristic polynomial of this matrix? }\\\ \begin{array}{cccccc} -a_{1} & a_{2} & a_{3} & \cdots & a_{n-1} & a_{n} \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ddots & 0 & 0 \\ -0 & 0 & 0 & \cdots & 1 & 0 \end{array} \end{array} $$

Prove that if \(A\) is normal, then the condition \(A B=B A\) implies that \(A^{*} B=B A^{*}\).

If \(U\) and \(V\) are unitary, does it follow that $$ \left[\begin{array}{ll} U & 0 \\ 0 & V \end{array}\right] $$ is unitary?

Prove that if \(x\) and \(y\) are points in \(\mathbb{C}^{n}\) having the same Euclidean norm, then there is a unitary matrix \(U\) such that \(U x=y\).

Let \(A=L U\), where \(L\) is unit lower triangular and \(U\) is upper triangular. Put \(B=U L\), and show that \(B\) and \(A\) have the same eigenvalues.

a. Prove that the set of all unitary matrices (of fixed order) is a multiplicative group. b. Prove that the set is closed under the operations \(A \rightarrow A^{T}, A^{*}\), and \(\bar{A}\).

Prove or disprove: If \(\left\\{x_{1}, x_{2}, \ldots, x_{k}\right\\}\) and \(\left\\{y_{1}, y_{2}, \ldots, y_{k}\right\\}\) are orthonormal sets in \(\mathbb{C}^{n}\), then there is a unitary matrix \(U\) such that \(U x_{i}=y_{i}\) for \(1 \leq i \leq k\).

Use Householder's algorithm to find the \(Q R\) -factorization of $$ \left[\begin{array}{rr} 0 & -4 \\ 0 & 0 \\ -5 & -2 \end{array}\right] $$