Multiplying and Factoring Matrices

Abstract

All of us learn and teach matrix multiplication using rows times columns. Those inner products are the entries of AB. But to go backward—to factor a matrix into triangular or orthogonal or diagonal matrices—outer products are much better. Now AB is the sum of columns of A times rows of B: rank one matrices. Our goal is to produce those columns and rows as simply as possible for A = LU (elimination) and A = CE (echelon form) and A = QR (Gram–Schmidt). Diagonalization by eigenvectors and by singular vectors is also expressed by columns times rows.