Chapter 4 - Canonical Forms Obtained Via Orthogonal Transformations

Abstract

This chapter presents two very important canonical forms: the Hessenberg form and the Real Schur Form (RSF) of a matrix can be obtained using orthogonal similarity transformations. Another important canonical form, known as the generalized real Schur form, can be obtained using orthogonal equivalence. These canonical forms form important tools in the development of numerically effective algorithms for control problems. The RSF of a matrix displays the eigenvalues of the matrix. The chapter presents a method, known as the QR iteration method, for computing the RSF of the matrix. The QR iteration method is nowadays a standard method for computing the eigenvalues of a matrix. The QR factorization is iterative in nature. Since the roots of a polynomial equation of degree higher than four cannot be found in a finite number of steps, any numerical method to compute the eigenvalues of a matrix of higher order than four, has to be iterative in nature. The chapter also explains the generalized Real Schur Form by QZ algorithm and the Golub-Kahan-Reinsch algorithm that is a standard computational algorithm for computing the SVD.