(a) Show that for all $u, v \in \mathbf{R}^p$, we have $\left(uu^T\right) \odot \left(vv^T\right) = (u \odot v)(u \odot v)^T$. (b) Let $A \in \operatorname{Sym}^+(p)$. We denote $\lambda_1, \ldots, \lambda_p$ the eigenvalues (with multiplicity) of $A$ and $\left(u_1, \ldots, u_p\right)$ an orthonormal family of associated eigenvectors. Show that $\lambda_k \geq 0$ for all $k \in \llbracket 1, p \rrbracket$ and that $A = \sum_{k=1}^{p} \lambda_k u_k u_k^T$. (c) Deduce that if $A, B \in \operatorname{Sym}^+(p)$ then $A \odot B \in \operatorname{Sym}^+(p)$.
(a) Show that for all $u, v \in \mathbf{R}^p$, we have $\left(uu^T\right) \odot \left(vv^T\right) = (u \odot v)(u \odot v)^T$.
(b) Let $A \in \operatorname{Sym}^+(p)$. We denote $\lambda_1, \ldots, \lambda_p$ the eigenvalues (with multiplicity) of $A$ and $\left(u_1, \ldots, u_p\right)$ an orthonormal family of associated eigenvectors. Show that $\lambda_k \geq 0$ for all $k \in \llbracket 1, p \rrbracket$ and that $A = \sum_{k=1}^{p} \lambda_k u_k u_k^T$.
(c) Deduce that if $A, B \in \operatorname{Sym}^+(p)$ then $A \odot B \in \operatorname{Sym}^+(p)$.