trace of $M \mapsto PM+MP$ with P a matrix of projection

Question

let $n \in \mathbb{N} \:\: n\geq2$ and $P \in \mathcal{M}_n(\mathbb{R})$ a matrix of projection

$\begin{array}{ccccc} Φ & : & \mathcal{M}_n(\mathbb{R}) & \to & \mathcal{M}_n(\mathbb{R}) \\ & & M & \mapsto & PM + MP \\ \end{array}$

Express $Tr(Φ)$ as a function of $Tr(P)$

Answer 1

Consider it in $\mathcal M_n(\mathbb C)$.

Suppose $P(e_{ij})=e_{ij}$ for $i,j\leq k$ and $P(e_{ij})=0$ for $i,j>k$.

$\text{Tr}(\Phi)=\sum_{ij} \langle \Phi(e_{ij}),e_{ij}\rangle=k^2=\text{Tr}(P)^2$