Piziak, R.; Odell, P. L. Full rank factorization of matrices. (English) Zbl 1006.15009 Math. Mag. 72, No. 3, 193-201 (1999). Summary: There are various useful ways to write a matrix as the product of two or three other matrices that have special properties. For example, today’s linear algebra texts relate Gaussian elimination to the LU factorization and the Gram-Schmidt process to the QR factorization. In this paper, we consider a factorization based on the rank of a matrix. Our purpose is to provide an integrated theoretical development of and setting for understanding a number of topics in linear algebra, such as the Moore-Penrose generalized inverse and the singular value decomposition. We make no claim to a practical tool for numerical computation – the rank of a very large matrix may be difficult to determine. However, we will describe two applications; one to the explicit computation of orthogonal projections, and the other to finding explicit matrices that diagonalize a given matrix. Cited in 23 Documents MSC: 15A23 Factorization of matrices 65F25 Orthogonalization in numerical linear algebra 65F20 Numerical solutions to overdetermined systems, pseudoinverses 15A03 Vector spaces, linear dependence, rank, lineability 15A21 Canonical forms, reductions, classification 65F05 Direct numerical methods for linear systems and matrix inversion Keywords:diagonalization; full rank factorization; Gaussian elimination; LU factorization; Gram-Schmidt process; QR factorization; Moore-Penrose generalized inverse; singular value decomposition; orthogonal projections PDFBibTeX XMLCite \textit{R. Piziak} and \textit{P. L. Odell}, Math. Mag. 72, No. 3, 193--201 (1999; Zbl 1006.15009) Full Text: DOI