Block Multiplication of Matrices

(We needed this to derive the conditional distribution of a multivariate Gaussian).

Consider a matrix product $AB$. Partition the two outer dimensions (i.e. the rows of $A$ and the columns of $B$) and the one inner dimension (i.e. the columns of $A$ and the rows of $B$) arbitrarily. This defines a “block decomposition” of the product $AB$ and of the factors $A, B$ such that the blocks of $AB$ are related to the blocks of $A$ and $B$ via the familiar formula for components of the product, i.e.

$(AB)_{m,n} = \sum_s A_{m,s} B_{s,n}$.

Pictorially, we have the following:

Arithmetically, this is easy to prove by considering the formula above for the components of the product. The partitioning of the outer dimensions comes for free, while the partitioning of the inner dimension just corresponds to partitioning the summation:

$(AB)_{m,n} = \sum_s A_{m,s} B_{s,n} = \sum_i \sum_{s_i \leq s \leq s_{i+1}} A_{m,s} B_{s,n}$.

Zooming out to a categorical level, we can see that there is nothing peculiar about this situation. If, in an additive category, we have three objects $X, Y, Z$ with biproduct decompositions, and a chain of morphisms:

$X \xrightarrow{\varphi_B} Y \xrightarrow{\varphi_A} Z$

then this “block decomposition of matrices” finds expression as a formula in $\text{End}(X, Z)$ using the injection and projection morphisms associated with each biproduct factor.