I was wondering what relations and similarities are between direct product for matrices and direct product for vector spaces? Or do they just unfortunately and somehow misleadingly happen to have the same name?
Note that the direct product for matrices is also called Kronecker product or tensor product of matrices.
I was wondering if there is a similar thing for matrices just as direct product for vector spaces? Direct sum for matrices seems to correspond to direct sum for vector spaces, instead of direct product for vector spaces. So I guess direct sum for matrices is not the answer?
Thanks to Arturo for his comment:
You can connect direct sums of matrices with direct sums of vector spaces in the following sense: if $A$ is an $n×m$ matrix and $B$ is a $p×q$ matrix, then $A⊕B$ is the block diagonal matrix that has upper left block $A$ and bottom right block $B$. Interpreting $A$ as a map $F_m→F_n$ and $B$ as a map $F_q→F_p$, then $A⊕B$ is the corresponding map $F_m⊕F_q→F_n⊕F_p$.
Since direct sum and direct product of vector spaces share so much similarity, why is it direct sum instead of direct product of vector spaces that the direct sum of matrices correspond to?