[Math] Is the set of all $n\times n$ matrices, such that for a fixed matrix B AB=BA, a subspace of the vector space of all $n\times n$ matrices

linear algebravector-spaces

Is the set of all $n\times n$ matrices, such that for any fixed matrix B AB=BA, a subspace of the vector space of all $n\times n$ matrices?

Alright, I understand the question and I know what I have to do, basically. I need to show that additive closure and multiplicative closure are satisfied. The problem is, I can't figure out how to do this generally. I tried playing around with $2\times 2$ matrices but that seemed like a dead end. Obviously two such matrices are the 0 matrix and the identity matrix, and those form a subspace, but that doesn't really tell me about all the matrices. Any ideas for how I should be tackling this? I feel like I'm not thinking generally enough.

Best Answer

Careful: when you say "multiplicative closure", you have to be clear, since when dealing with $n\times n$ matrices there is a "multiplication" that has nothing to do with the vector space structure (the multiplication of matrices). It is clearer if you refer to it as the "scalar multiplication".

So, fix the matrix $B$. You need to show that:

  1. There is at least one matrix $A$ such that $AB=BA$;
  2. If $A_1$ and $A_2$ are two matrices such that each of them commutes with $B$, then $A_1+A_2$ also commutes with $B$ (closure of your set under vector addition).
  3. If $A$ is a matrix that commutes with $B$, and $k$ is any scalar, then $kA$ also commutes with $B$ (closure of your set under scalar multiplication).

So, with that in mind:

  1. Is there a matrix that necessarily commutes with $B$? (If this thing is really going to be a subspace, it better have what "vector" [i.e., matrix] in it for sure? Try that matrix).

  2. Suppose $A_1$ and $A_2$ both commute with $B$. That is, $A_1B=BA_1$ and $A_2B=BA_2$. You want to show that $(A_1+A_2)$ also commutes with $B$. That is, you want to show that $$(A_1+A_2)B = B(A_1+A_2).$$ Of course, you'll want to use the fact that each of $A_1$ and $A_2$ commutes with $B$, and perhaps some properties you know about matrix multiplication. Is there some property of matrix multiplication that would let you relate $(A_1+A_2)B$ with the products you do know something about, namely $A_1B$ and $A_2B$?

  3. Suppose $A$ commutes with $B$, and $AB=BA$. Let $k$ be a scalar. You want to show that $kA$ also commutes with $B$: that is, you need to prove that $$(kA)B = B(kA).$$ Again, is there some property of matrix multiplication that you know and that might help here?

And if you establish these three, you're done: the set in question is a subspace!

Related Question