rank factorization

A matrix of rank r contains only r independent columns; everything else is a combination of them. Rank factorization makes this concrete: any m-by-n matrix A of rank r can be written A = C F, where C is m-by-r with full column rank (its columns are a basis for the column space of A) and F is r-by-n with full row rank (its rows tell you how to mix C's columns to rebuild every column of A).

It is the leanest exact representation of A. Instead of storing m*n numbers, you store m*r + r*n, which is a huge saving when r is small. One easy way to build it: row reduce A, take the pivot columns of the original A as C, and take the nonzero rows of the reduced row echelon form as F.

Conceptually it certifies the rank. The very existence of a product C F with C full-column-rank and F full-row-rank forces the inner dimension to equal rank(A); you cannot squeeze A through a thinner middle. This is the algebraic heart of why row rank equals column rank.

It is also the skeleton behind low-rank approximation and the pseudoinverse. The SVD gives a particularly clean, orthonormal rank factorization, and truncating it to the top r singular triples is the best rank-r approximation. Plain rank factorization is the exact, non-unique version; the SVD is its canonical, optimal refinement.