There is a group action $$GL(N, \mathbb{R}) \times GL(N, \mathbb{R}) \times Mat_N (\mathbb{R}) \rightarrow Mat_N (\mathbb{R})$$ $$(A, B, X) \mapsto AXB^{-1} $$
I would like to find stabilizers for this action. I was able to find the orbits of action $$GX = \left\lbrace Y \in Mat_N (\mathbb{R}): rk(Y) = rk(X) \right\rbrace$$
In order to find stabilizers, we must find all solutions of equation $$AXB^{-1} = X$$ for A, B. I know that stabilizer is Lie group and Lie algebra for this stabilizer is $$\mathfrak{g}_X = \left\lbrace (A, B) \in Mat_N (\mathbb{R}) \times Mat_N (\mathbb{R}) : AX - XB = 0 \right\rbrace$$ My question is whether we can more accurately indicate the stabilizer $G_X$?
Thank you so much