Due date: Monday 14. October 2024, 10 AM.
Exercise 1
Compute the Cholesky decomposition of the symmetric positive definite matrix \[\mathbf{A}=\begin{pmatrix} 36 & 6 & 10 \\ 6 & 10 & 0 \\ 10 & 0 & 10 \end{pmatrix}.\]
Exercise 2
Let \(V=M_{n,n}(\mathbb{R})\) and consider \(\langle \cdot{,}\cdot\rangle: V \times V \to \mathbb{R}\) defined by the rule \(\langle \mathbf{A},\mathbf{B}\rangle=\operatorname{Tr}(\mathbf{A}^T\mathbf{B})\) for all \(\mathbf{A},\mathbf{B}\in V.\)
Show that \(\langle \cdot{,}\cdot\rangle\) is an inner product;
find an orthonormal basis for \((V,\langle \cdot{,}\cdot\rangle).\)
Exercise 3
Let \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) and consider the linear maps \(f=f_\mathbf{A},f^T=f_{\mathbf{A}^T}:\mathbb{E}^n\to\mathbb{E}^n.\) Show that \(\operatorname{Ker}(f) \perp \mathrm{Im}(f^T)\) and \(\mathbb E^n = \operatorname{Ker}(f) \oplus \mathrm{Im}(f^T).\)
Exercise 4
Let \(r_1,\ldots,r_n\in \mathbb{R}\) be non-negative numbers such that \(r_1+r_2+\cdots + r_n =1.\) Show that \[\sum_{k=1}^n \sqrt{r_k} \leqslant \sqrt n.\]
Let \(\mathbf{A}\in M_{n,n}(\mathbb{R}).\) Show that \(\operatorname{Tr}(\mathbf{A})^2 \leqslant n \operatorname{Tr}(\mathbf{A}^T\mathbf{A}).\)