On several occasions, I have heard that matrix multiplication is commutative for square matrices $A$ and $B$ when they represent linear transformations. Is this true? I know that in general $AB$ is not $BA$ for some matrices $A$ and $B$.
[Math] Is matrix multiplication commutative for square matrices representing linear transforms
linear algebramatricesnumerical linear algebra
Best Answer
All $n \times n$ square matrices (say, over the field $\mathbb{F}$) can be regarded as linear transformations of an $n$-dimensional vector space $\mathbb{V}$ over $\mathbb{F}$ in a given basis, which allows us to identify $\mathbb{V} \cong \mathbb{F}^n$. In particular, as you note in general $AB$ and $BA$ differ, and hence the answer is no.