Let A be a fixed n x n matrix and let S be the set of all n

Let A be a fixed n x n matrix and let S be the set of all n x n matrices that commute with A. Show that S is a subspace of the vector space of all n x n matrices.

Solution

You might be confused because of the notation. I am guessing that A is a fixed nxn matrix. And we have the set of all nxn matrices B that satisfy AB = BA. That is cool--- but this set should not be called B; it should be called something else, say, S.

S = {the set of B for which AB = BA}.

The reason we need another letter for it is that it is potentially quite confusing to use the name of a set in the definition of a set. (As an example: \"let x be the set of all solutions x to x^2 + 2x + 1 = 0\" is rough because x is both a number and a set of numbers; in the phrasing given, B is both a set of matrices and a matrix. It is no wonder you were confused.)

You could phrase this slightly differently without symbols: fix an nxn matrix A, and let S denote the set of nxn matrices that commute with A. Is S a subset of the space of all nxn matrices (which you presumably already know is a vector space)? The answer is yes. This is done by showing that S is closed under addition and scalar multiplication, or in words, that if two matrices commute with A, so does their sum, and if a matrix commutes with A, then so does any scalar multiple of it. In symbols:

If B and B\' are two matrices satisfying AB=BA and AB\' = B\'A then

A(B+B\') = AB + AB\' = BA + B\'A = (B+B\')A,

and if k is a scalar,

A(kB) = k(AB) = k (BA) = (kB) A.

This shows that S is closed under addition and scalar multiplication: so it is a subspace of the space of all nxn matrices.