Prove continuity of application from $M_n(R)$ to $L(M_n(R))$

Question

Let $A \in M_n(R)$ and $\rho_A : M_n(R) \rightarrow M_n(R)$ with $\rho_A(M) = A^{T}MA$.

How can I properly prove that $\rho : A \mapsto \rho_A$ is a continuous application from $M_n(R)$ to $L(M_n(R))$.

Answer 1

Let $||\cdot||$ be a matrix norm on $M_{n}(\mathbb{R})$.

For every matrices $A$, $B$ and $M$ we have

$\rho_A(M) - \rho_B(M) = (A - B)^T M A + B^T M (A - B)$ and so $\left\|\rho_A(M) - \rho_B(M)\right\| \le \left\|(A - B)^T\right\| \left\|M\right\|\left\| A\right\| + \left\|B^T\right\|\left\| M \right\| \left\|(A - B)\right\|$ dividing by $\left\|M\right\|$ and taking the supremum we have :

$\left\|\rho_A - \rho_B\right\| \le \left\|(A - B)^T\right\|\left\| A\right\| + \left\|B^T\right\|\left\|(A - B)\right\|$. Finally using the continuity of $M \mapsto M^T$ we can conclude the continuity of $\rho$.