91Ó°ÊÓ

If \(A\) is Hermitian, what is the relationship between its eigenvalues and its singular values?

Prove that if one column in the matrix \(A\), say the \(j\) th column, satisfies \(a_{i j}=0\) for \(i \neq j\), then \(a_{j j}\) is an eigenvalue of \(A\).

For fixed \(u\) and \(x\), what value of \(t\) makes the expression \(\|u-t x\|_{2}\) a minimum? (The answer is to be valid in the complex case.)

Let the eigenvalues of \(A\) satisfy \(\lambda_{1}>\lambda_{2}>\cdots>\lambda_{n}(\) all real, but not necessarily positive). What value of the parameter \(\beta\) should be used in order for the power method to converge most rapidly to \(\lambda_{1}\) when applied to \(A+\beta I\) ?

A normal matrix is one that commutes with its adjoint: \(A A^{*}=A^{*} A\). Prove that if \(A\) is normal, then so is \(A-\lambda I\) for any scalar \(\lambda\).

Prove that the matrix having elements \(\left\langle x_{i}, y_{j}\right\rangle\) is unitary if \(\left\\{x_{1}, x_{2}, \ldots, x_{n}\right\\}\) and \(\left\\{y_{1}, y_{2}\right.\), \(\left.\ldots, y_{n}\right\\}\) are orthonormal bases for \(\mathbb{C}^{n} .\)

Suppose that \(A\) is normal and that \(x\) and \(y\) are eigenvectors of \(A\) corresponding to different eigenvalues. Prove that \(x^{*} y=0\).

Prove that \(I-A B\) has the same eigenvalues as \(I-B A\), if either \(A\) or \(B\) is nonsingular.

A matrix \(A\) such that \(A^{2}=I\) is called an involution or is said to be involutory. Find the necessary and sufficient conditions on \(u\) and \(v\) in order that \(I-u v^{*}\) be an involution.

(Continuation) Let \(v^{*} v=2\). Show that the partitioned matrix $$ \left[\begin{array}{rr} I & 0 \\ 0 & I-v v^{*} \end{array}\right] $$ is an involution.