Set of matrices invariant under multiplication by an external matrix

Question

I'm looking for the set of matrices (name and properties if possible) which is closed under multiplication by another matrix.

Example:

a set of matrices S=$\{\mathbf{A}_1,\mathbf{A}_2,\ldots,\mathbf{A}_n\}$ and an external matrix $\mathbf{B}$ such that $\mathbf{B}\mathbf{A}_i=\mathbf{A}_j \quad\forall\, i,j \in \{1\ldots n\},~i \neq j$.

Answer 1

To amplify on Servaes's comment:

For any invertible matrix $B$, consider the following two sets:

$$ S_1 = \{0\}\\ S_2 = \{ \ldots, B^{-2}, B^{-1}, I, B, B^2, \ldots\}\\ $$

For each of these, we have $B\cdot S_i = S_i$. So the idea that there's such a thing as "the set" with this property is ill-founded.

For $B$ not invertible, $S_1$ is still a left-invariant set; there may be no others, however (as can be seen in the case where $B = 0$).