Not every matrix on V ⊗ W can be written as a tensor product of a matrix on V and another on W.

I am reading some notes about tensor product of vector spaces (those in here) where the following sentence can be found:

Note that not every matrix on V ⊗ W can be written as a tensor product of a matrix on V and another on W.

Matrices in V ⊗ W are defined as a Kronecker product. So, how could this claim be possible? Has it to do with the problem of splitting the Kronecker product of two matrices?

More generally: does the general picture changes if we consider V and W infinite-dimensional vector spaces?

2. Consider a $$4 \times 4$$ matrix $$C$$ with only one zero entry. If you try to write $$C = A \otimes B$$, either none of the $$A,B$$ entries is equal to zero and then $$C$$ will have no zero entry. Or they have at least one, and then $$C$$ will have at least four zero entries.