Subspaces containing matrices that commute with a given matrix

• Sep 27th 2011, 07:35 AM
charmedquark
Subspaces containing matrices that commute with a given matrix
Problem:
Determine the subspace of R(2 by 2) consisting of all matrices that commute with the given matrix:
(1 0 ) =B
(0 -1)
Attempt at sol'n:
I wanted a general 2 by 2 matrix A so that AB=BA. Multiplying out component wise, I think that it must be the case that a11=a11, a12=-a12=0, -a21=a21=0, and -a22=-a22. So I think that
(1 0 ) (a11 0 ) = (a11 0 ) (1 0 )
(0 -1) (0 -a22) (0 -a22) (0 -1)

And I already know that if S is a subspace of R(n by n) where S is the set of all matrices that commute with a fixed vector in R(n by n) that S is a subspace of R(n by n).
So is the above sufficient or do I need to do something entirely different?
• Sep 27th 2011, 07:43 AM
Ackbeet
Re: Subspaces containing matrices that commute with a given matrix
You're pretty much there, I'd say. The subspace you're seeking is the subspace of diagonal matrices. All you need check is closure under scalar multiplication and vector addition, which is easy.
• Sep 27th 2011, 07:52 AM
charmedquark
Re: Subspaces containing matrices that commute with a given matrix
Well I know from the fact that S is a subspace of R(n by n) that my particular example is closed under addition and multiplication (right?). But I'm not sure how to write my answer in set builder notation. For my particular S,

S={A such that ?}
• Sep 27th 2011, 08:03 AM
Ackbeet
Re: Subspaces containing matrices that commute with a given matrix
You can do the commuting-with-fixed-vector approach, or the approach I mentioned. As for writing your final subspace in set builder notation, how about this:

$\displaystyle S=\left\{M\in\mathbb{R}^{2\times 2}\Big|M=\begin{bmatrix}x&0\\ 0&y\end{bmatrix}\;\text{for some}\; x,y\in\mathbb{R}\right\}.$
• Sep 27th 2011, 08:06 AM
charmedquark
Re: Subspaces containing matrices that commute with a given matrix
I think I understand. Thank you for your time.
• Sep 27th 2011, 08:07 AM
Ackbeet
Re: Subspaces containing matrices that commute with a given matrix
You're very welcome. Have a good one!