Due date: Monday 18. November 2024, 10 AM.
Exercise 1
Let \(f : V \to V\) be a normal endomorphism of the finite dimensional unitary space \((V,\langle \cdot{,}\cdot\rangle).\) Show that \(\operatorname{Ker}f = (\mathrm{Im}f)^{\perp}.\)
Exercise 2
Let \[\mathbf{A}=\begin{pmatrix} 0 & 1 && \\ & 0 & 1 & \\ && \ddots & 1 \\ 1 &&& 0 \end{pmatrix}\] Show that \(f_\mathbf{A}: \mathbb{C}^n \to \mathbb{C}^n\) is a normal endomorphism with respect to the standard Hermitian inner product. Find a ordered basis of \(\mathbb{C}^n\) consisting of eigenvectors of \(f_\mathbf{A}.\)
Exercise 3
Let \(\mathbf{A}\in M_{n,n}(\mathbb{C})\) be normal and such that \(\mathbf{A}^2=\mathbf{A}.\) Show that \(\mathbf{A}\) is Hermitian.
Exercise 4
Show that every normal matrix \(\mathbf{A}\in M_{2,2}(\mathbb{R})\) is either symmetric or of the form \[\mathbf{A}= a \mathbf{1}_2 + \mathbf{B},\] where \(\mathbf{B}\) is antisymmetric and \(a\in \mathbb{R}.\)
Show that the set of normal matrices does not form a subspace of \(M_{2,2}(\mathbb{C}).\)