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Full rank factorization of matrices. (English) Zbl 1006.15009

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.

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
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