This paper presents a mathematical framework to exploit the semantic properties of matrix operations in loop-based numerical codes. The heart of this framework is an algebraic language called the Abstract Ma-trix Form which a compiler can use to reason about matrix computations in terms of loop nests, high-level matrix operations, and intermediate forms. We demonstrate how this framework may be used to detect and exploit matrix products in loop-based languages such as FORTRAN and MATLAB, and discuss the resulting performance benefits.