Do tensors in component form commute?

Question

For arbitrary tensors $\mathbf{A}$ and $\mathbf{B}$, does this commutativity relation for the tensor components always hold? Does it hold for dummy indices?

\begin{equation} {A^{\mu_1...\mu_n}}_{\nu_1...\nu_m}{B^{\alpha_1...\alpha_j}}_{\beta_1...\beta_k} = {B^{\alpha_1...\alpha_j}}_{\beta_1...\beta_k}{A^{\mu_1...\mu_n}}_{\nu_1...\nu_m} \end{equation}

If not, how would I prove that? If it holds for some tensors, how would I check? If always, is there an easy proof of this?

I’ve tried to find this elsewhere but am either searching for the wrong terms or not finding anything.

Answer 1

These $A^{\mu_1 ... \mu_n}_{\nu_1 ... \nu_m}$ and $B^{\alpha_1 ... \alpha_j}_{\beta_1 ... \beta_k}$ are just numbers, so they commute, as elements of your field. It's the same as writing $\vec{y} = M\vec{x} \rightarrow y_i = \sum M_{ij}x_j = \sum x_j M_{ij}$.

Answer 2

The product $A^{\mu_1...\mu_n}_{\nu_1...\nu_m}B^{\alpha_1...\alpha_j}_{\beta_1...\beta_k}$ is itself a tensor of rank $(n+m+j+k)$ and in no way is a scalar. But their product is commutative because the resulting tensor product has the same contravariant and covariant indices. Also this conclusion doesn't depend on the choice of coordinates since tensor equations are in general covariant.