91Ó°ÊÓ

Show that if a matrix \(A\) is both triangular and unitary, then it is diagonal.

The Pythagorean theorem asserts that for a set of \(n\) orthogonal vectors \(\left\\{x_{i}\right\\}\) \\[ \left\|\sum_{i=1}^{n} x_{i}\right\|^{2}=\sum_{i=1}^{n}\left\|x_{i}\right\|^{2} \\] (a) Prove this in the case \(n=2\) by an explicit computation of \(\left\|x_{1}+x_{2}\right\|^{2}\) (b) Show that this computation also establishes the general case, by induction.

What can be said about the eigenvalues of a unitary matrix?

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