Problem 11
Show that if \(\mathbf{A}\) is invertible, then the eigenvectors of \(\mathbf{A}^{-1}\) are the same as those of \(\mathbf{A}\) and the eigenvalues of \(\mathbf{A}^{-1}\) are the reciprocals of those of \(\mathbf{A}\).
Problem 12
Find all eigenvalues and eigenvectors of the following matrices: \(\mathrm{A}_{1}=\left(\begin{array}{ll}1 & 1 \\ 0 & i\end{array}\right) \quad \mathrm{B}_{1}=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right) \quad \mathrm{C}_{1}=\left(\begin{array}{ccc}2 & -2 & -1 \\ -1 & 3 & 1 \\ 2 & -4 & -1\end{array}\right)\) \(\mathrm{A}_{2}=\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right) \quad \mathrm{B}_{2}=\left(\begin{array}{lll}1 & 1 & 0 \\\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right) \quad \mathrm{C}_{2}=\left(\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right)\) \(\mathrm{A}_{3}=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right) \quad \mathrm{B}_{3}=\left(\begin{array}{lll}1 & 1 & 1 \\\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right) \quad \mathrm{C}_{3}=\left(\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right)\)
Problem 13
Show that a \(2 \times 2\) rotation matrix does not have a real eigenvalue (and, therefore, eigenvector) when the rotation angle is not an integer multiple of \(\pi\). What is the physical interpretation of this?
Problem 14
Three equal point masses are located at \((a, a, 0),(a, 0, a)\), and \((0, a, a) .\) Find the moment of inertia matrix as well as its eigenvalues and the corresponding eigenvectors.
Problem 15
Consider \(\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \in \mathbb{C}^{n}\) and define \(\mathbf{E}_{i j}\) as the operator that interchanges \(\alpha_{i}\) and \(\alpha_{j} .\) Find the eigenvalues of this operator.
Problem 16
Find the eigenvalues and eigenvectors of the operator \(-i d / d x\) acting in the vector space of differentiable functions \(\mathcal{C}^{1}(-\infty, \infty)\).
Problem 17
Show that a hermitian operator is positive iff its eigenvalues are positive.
Problem 18
Show that \(\|\mathbf{A} x\|=\left\|\mathbf{A}^{\dagger} x\right\|\) if and only if \(\mathbf{A}\) is normal.
Problem 19
What are the spectral decompositions of \(\mathbf{A}^{\dagger}, \mathbf{A}^{-1}\), and \(\mathbf{A A}^{\dagger}\) for an invertible normal operator \(\mathbf{A}\) ?
Problem 20
Consider the matrix $$A=\left(\begin{array}{cc} 2 & 1+i \\\1-i & 3\end{array}\right) .$$ (a) Find the eigenvalues and the orthonormal eigenvectors of A. (b) Calculate the projection operators (matrices) \(\mathrm{P}_{1}\) and \(\mathrm{P}_{2}\) and verify that \(\sum_{i} P_{i}=1\) and \(\sum_{i} \lambda_{i} P_{i}=\mathrm{A} .\) (c) Find the matrices \(\sqrt{\mathrm{A}}, \sin (\theta \mathrm{A})\), and \(\cos (\theta \mathrm{A})\) and show directly that $$\sin ^{2}(\theta \mathrm{A})+\cos ^{2}(\theta \mathrm{A})=1$$ (d) Is A invertible? If so, find \(A^{-1}\) using spectral decomposition of A.
