For all of the matrices \(\mathbf{A}\) below, decide whether the matrix is diagonalisable and if so, find a matrix \(\mathbf{C}\) such that \(\mathbf{C}\mathbf{A}\mathbf{C}^{-1}\) is diagonal. \[\text{(a)}\;\; \begin{pmatrix} -2 & 2 \\ -2 & 3 \end{pmatrix}, \quad \text{(b)}\;\;\begin{pmatrix} 1 & \mathrm{i}\\ -\mathrm{i}& 1 \end{pmatrix}, \quad \text{(c)}\;\; \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}, \quad \text{(d)}\;\; \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.\]

Let \(g : V \to V\) be an endomorphism of a finite dimensional \(\mathbb{K}\)-vector space \(V.\)

1. Suppose \(g\) has two linearly independent eigenvectors with respect to the same eigenvalue \(\lambda.\) Does this imply that the algebraic multiplicity of \(\lambda\) is bigger than one?

2. Suppose \(\lambda\) is an eigenvalue of \(g\) with algebraic multiplicity bigger than one. Does \(g\) then admit two linearly independent eigenvectors with respect to the eigenvalue \(\lambda\)?

Let \(\theta\in \mathbb{R}\) and consider \[\mathbf{A}=\begin{pmatrix} \cos\theta & - \sin \theta \\ \sin \theta & \cos \theta\end{pmatrix} \in M_{2,2}(\mathbb{K})\] Is \(\mathbf{A}\) conjugate to a diagonal matrix? If so, find a matrix \(\mathbf{C}\in M_{2,2}(\mathbb{K})\) such that \(\mathbf{C}\mathbf{A}\mathbf{C}^{-1}\) is diagonal. Consider the cases \(\mathbb{K}=\mathbb{R}\) and \(\mathbb{K}=\mathbb{C}.\)

Let \(g : V \to V\) be an endomorphism of an \(n\) dimensional \(\mathbb{K}\)-vector space \(V\) with eigenvalues \(\{\lambda_1,\ldots,\lambda_n\}.\)

1. Show that \(\det g=\prod_{i=1}^n \lambda_i\) and \(\operatorname{Tr} g = \sum_{i=1}^n \lambda_i.\)

2. Find expressions for the remaining coefficients of the characteristic polynomial.

(\(\heartsuit\)). Consider the endomorphism \(L_\mathbf{A}\) of \(M_{n,n}(\mathbb{K})\) obtained by left multiplication with \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) that is, \[L_\mathbf{A}: M_{n,n}(\mathbb{K}) \to M_{n,n}(\mathbb{K}), \qquad \mathbf{B}\mapsto \mathbf{A}\mathbf{B}.\] Compute \(\operatorname{Tr} L_\mathbf{A}\) and \(\det L_\mathbf{A}.\)