Chapter 3 - Direct algorithms of decompositions of matrices by orthogonal transformations

Abstract

Chapter 3, Direct Algorithms of Decompositions of Matrices by Orthogonal Transformations, addresses the decomposition of a general matrix and some special matrices. The decomposition of a matrix is itself an important matrix computation problem, but also the foundation of other matrix algorithms. The chapter starts with the Householder and Givens elimination matrices. Then it applies the two elimination matrices to build 2 flavors of QR algorithm. Further it presents the third flavor of QR algorithm that is Gram-Schimdt algorithm. Following the QR algorithm, which is to reduce a general matrix to an upper triangle matrix R by an orthogonal matrix Q, QLZ and QBZ algorithms are presented, which are to reduce a general matrix to a lower square triangle (L) or bi-diagonal (B) by orthogonal matrices Q and Z. For a square matrix, an orthogonal similarity transformation is possible. A QHQt algorithm is presented, which is to reduce a square matrix to a Hessenberg matrix. If a matrix is symmetric, to reduce it to a symmetric tri-diagonal is possible. The algorithm presented is called QTQt.