This chapter considers likely the most important operation in system theory: inner–outer and its dual, outer–inner factorization. These factorizations play a different role than the previously treated external or coprime factorizations, in that they characterize properties of the inverse or pseudo-inverse of the system under consideration, rather than the system itself. Important is that such factorizations are computed on the state-space representation of the original, that is, the original data. Inner–outer (or outer–inner) factorization is nothing but recursive “QR factorization,” as was already observed in our motivational Chapter 2, and outer–inner is recursive “LQ factorization,” in the somewhat unorthodox terminology used in this book for consistency reasons: QR for “orthogonal Q with a right factor R? and LQ for a “left factor” L with orthogonal Q?. These types of factorizations play the central role in a variety of applications (e.g., optimal tracking, state estimation, system pseudo-inversion, and spectral factorization) to be treated in the following chapters. We conclude the chapter showing how the time-variant, linear results generalize to the nonlinear case.

Chapter 7: Unitary matrices play important roles in theory and computation. The adjoint of a unitary matrix is its inverse, so unitary matrices are easy to invert. They preserve lengths and angles, and have remarkable stability properties in many numerical algorithms. In this chapter, we explore the properties of unitary matrices and present several special cases. We derive an explicit formula for a unitary matrix whose first column is given. We give a constructive proof of the QR factorization and show that every square complex matrix is unitarily similar to an upper Hessenberg matrix.

We study the solution of overdetermined systems of equations. Introduce weak, and in particular least squares solutions. For full rank systems, we show existence and uniqueness via the normal equations. We introduce projection matrices and the QR factorization. We discuss the computation of the QR factorization with the help of Householder reflectors. For rank defficient systems we prove the existence and uniqueness of a minimal norm least squares solution. We introduce the Moore-Penrose pseudoinverse, show how it relates to the SVD, and how it can be used to solve rank defficient systems.