The product of matrix subspaces

In factoring matrices into the product of two matrices, operations are typically performed with elements restricted to matrix subspaces. Such modest structural assumptions are realistic, for example, in large scale computations. This paper is concerned with analyzing associated matrix geometries. Curvature of the product of two matrix subspaces is assessed. The case of vanishing curvature gives rise to the notion of factorizable matrix subspace. This can be regarded as an analogue of the internal Zappa-Szép product in group theory. Interpreted in this way, several classical instances are encompassed by this structure. The Craig-Sakamoto theorem fits naturally into this framework.