As you can see from Kinds of Classes, many classes of the Linear Algebra Module involve either factorizations or decompositions. Although the terms are similar, they are not interchangeable. Both factorizations and decompositions are representations of a matrix as a product of other matrices. For example, A=LU is a factorization of the matrix A into factors L and U. The factors are typically of some special form that provides insight into the properties of A, or simplifies certain types of computations. Both L and U are triangular, which simplifies solving systems of equations.

The distinction between a factorization and a decomposition is that a factorization is designed so that you need never examine the factors, or even know their forms. In contrast, you are often interested in the specific components of decompositions. In an object-oriented library, this distinction can be enforced. Factorization classes allow no public access to the contents of the factorizations, while decompositions do allow public access.