This chapter contains topics related to matrices with special structures that arise in many applications. It discusses companion matrices that are a classic linear algebra topic. It constructs circulant matrices from a particular companion matrix and describes their signal processing applications. It discusses the closely related family of Toeplitz matrices. It describes the power iteration that is used later in the chapter for Markov chains. It discusses nonnegative matrices and their relationships to graphs, leading to the analysis of Markov chains. The chapter ends with two applications: Google’s PageRank method and spectral clustering using graph Laplacians.

We present several methods for approximating the spectrum of a matrix. We start by providing coarse estimates using so-called Gershgoring disks. The stability, via the Bauer-Fike theorem is then analyzed. For Hermitian matrices we then present and analyze the power iteration method and its variants. The reduction to Hessenberg form, and the QR algorithm are then presented and analyzed. The chapter then concludes with a discussion of the Golub-Kahan algorithm to compute the SVD.