91Ó°ÊÓ

If \(A\) is a complex square matrix, show that \(\operatorname{Det}(\bar{A})=\operatorname{Det}(A)\). Conelude that the absolute value of the determinant of a complex unitary matrix is equal to \(1 .\)

Let \(V\) be a finite dimensional space over the field \(K\), with a non- degenerate scalar product. Let \(A: V \rightarrow V\) be a linear map. Show that the image of \(^{t} A\) is the orthogonal space to the kernel of \(A\).

Let \(A\) be a diagonal complex unitary matrix. Show that each diagonal element has absolute value 1 , and hence is of type \(e^{i \theta}\), with real \(\theta\). The following exercises describe various properties of real unitary maps of the plane \(\mathbf{R}^{2}\).

Let \(A\) be a hermitian matrix. Show that \(^{t} A\) and \(\bar{A}\) are hermitian. If \(A\) is invertible, show that \(A^{-1}\) is hermitian.