bilinear form $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$, find ortogonal subspaces, that satisfy... Define $F$ as bilinear form $M_n(\mathbb{R}) \text{ x } M_n(\mathbb{R}) \rightarrow \mathbb{R}$ 
$F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$
Prove, that $F$ is represented by an invertible matrix on the space of matrices with trace equal to $0$
Find ortogonal subspaces $M_1, M_2, M_3$, that satisfy $M_n = M_1 \oplus M_2 \oplus M_3$ and $F$ is positive-definite on $M_1$, negative-definite on $M_2$ and constantly equal $0$ on $M_3$.
This is an extended level task from Linear Algebra course. I'm courious, how to solve it. 
I've tried to firstly find all matrices with  trace equal to $0$, then matrices for which $F(A,A)$ is always $> 0$. But I don't know how to do it. 
Thanks in advance for explaining! 
 A: The key thing to remember is that $\langle A,B \rangle : = \operatorname{Tr}(A^T B)$ defines an inner product on the vector space $M_n(\mathbb{R})$. Hence, in particular, there exists a unique linear transformation $\mathcal{F} : M_n(\mathbb{R}) \to M_n(\mathbb{R})$ such that
$$
 \forall A, B \in M_n(\mathbb{R}), \quad F(A,B) = \langle A, \mathcal{F}(B) \rangle;
$$
since $F$ is symmetric, it follows that $\mathcal{F}$ is self-adjoint, so that you can profitably apply the spectral theorem to it. Indeed, since
$$
 \operatorname{Tr}(AB) = \operatorname{Tr}(B^T A^T) = \operatorname{Tr}{A^T B^T} = \langle A, B^T \rangle,
$$
it follows that
$$
 F(A,B) = n\operatorname{Tr}(AB) - \operatorname{Tr}(A) \operatorname{Tr}(B) = n \operatorname{Tr}(A^T B^T) - \operatorname{Tr}(B)\operatorname{Tr}(A^T 1_n)\\
= \operatorname{Tr}(A^T (nB^T - \operatorname{Tr}(B)1_n)) = \langle A, nB^T - \operatorname{Tr}(B)1_n\rangle
$$
and hence that
$$
 \mathcal{F}(B) = nB^T - \operatorname{Tr}(B)1_n.
$$
So:


*

*$\operatorname{Tr}(\mathcal{F}(B)) = ?$

*What is the restriction of $\mathcal{F}$ to the subspace $\mathfrak{sl}(n,\mathbb{R}) \subset M_n(\mathbb{R})$ of matrices with vanishing trace?

*Where is $\mathcal{F}$ positive definite, where is it negative definite, and where does it vanish?

