Chapter 9 - SPECIAL MATRICES

Abstract

This chapter focuses on special matrices. The eigenvalues and the eigenvectors—including the generalized eigenvectors—of a matrix serve to characterize the matrix almost completely. With a knowledge of these properties, one can calculate the determinant of the matrix and, hence, determine its invertibility, find its Jordan canonical form, and compute functions of the matrix. There are certain classes of matrices, for example, real symmetric matrices, whose structures are such that their eigenvalues and eigenvectors are particularly simple. A nonzero vector is said to be normalized if it is divided by its magnitude. The chapter discusses orthonormal vectors. A set of vectors is called an orthogonal set if each vector in the set is orthogonal to every other vector in the set. A set of vectors is orthonormal if it is an orthogonal set having the property that every vector is a unit vector—a vector of magnitude 1.