Chapter 2 - Direct algorithms of decompositions of matrices by non-orthogonal transformations

Abstract

Chapter 2, Direct Algorithms of Decompositions of Matrices by Non-orthogonal Transformations, addresses the decompositions of a general matrix and some special matrices. It starts with Gauss elimination matrix and its numerical conditioning formula. Then it builds the LU decomposition of a general matrix, which is to factorize a general matrix to a lower (L) and upper (U) triangular matrices. Further the U factor is decomposed as a diagonal (D) and another upper triangular matrix (U) with 1s on its diagonal. For a symmetric matrix, it includes 3 congruent transformation algorithms, diagonal reduction (LDLt), tri-diagonal reduction (LTLt) and block diagonal reduction (LBLt). For a symmetric positive matrix, it has the Cholesky decomposition (LLt). Further a so-called modified Cholesky decomposition (xLLt) is presented for a symmetric positive indefinite matrix. For a square matrix, it presents 2 similarity transformation algorithms, Hessenberg reduction (LHLi) and tri-diagonal reduction (GTGi). Finally it builds an algorithm that simultaneously reduce a symmetric matrix to a tri-diagonal and another symmetric matrix to a diagonal of +/- 1s and 0s by similarity transformations. All the algorithms are presented in more than one forms, from which the inter-relations of different forms of algorithms can be understood.