A confusion on terminologies: "direct product" and "tensor product" of matrices

Question

In "Matrix Analysis & Applied Linear Algebra" by Meyers, the "direct product" of matrices is a synonym for the Kronecker product or the tensor product. I think the same terminologies are used in many quantum mechanics books. However, if I consider the vector space of matrices, the terminology "direct product" has a contradiction with usual usage. As far as I know the direct product of two vector spaces is a synonym of the direct sum (for example, here is Lang's definition). This usage of the term "direct product" is not the same with the usage by Meyers if we consider the vector space of matrices. Is this usage by Meyers usual for the space of matrices? Or is the term usually a synonym for the tensor product?

Answer 1

Note that the objects which form the factors are different in both cases you cite for the term direct product.

In the first case, the factors are indivial matrices, so e. g. for $A, B \in \mathrm{Mat}_n(k)$, we form $$ A \otimes B \in \mathrm{Mat}_{n^2}(k) $$ so this product maps matrices to matrices.

In the second case, the factors are non individual matrices, but whole vector spaces (possibly of matrices, but whole spaces of them). E. g. we can form the vector space $$ \mathrm{Mat}_n(k) \times \mathrm{Mat}_n(k) $$ as a direct product.

Hence, confusion is not likely, and as both usages of the term are common, that's a good thing.