Chapter 10 - Special Matrices

Abstract

This chapter describes special matrices of computational mathematics. It is apparent that the eigenvalues and the eigenvectors of a matrix serve to characterize the matrix almost completely. With them, one can calculate the determinant of the matrix, determine its invertibility, find its Jordan canonical form, and compute functions of the matrix. On the other hand, it is impossible to obtain information of this sort and numerical methods must be employed for most matrices. However, there are certain classes of matrices whose structures are such that their eigenvalues and eigenvectors are particularly simple; these matrices are called special matrices. The eigenvalues of a real symmetric matrix are real; however, the eigenvectors can always choose to be real. A real symmetric matrix is diagonalizable. A real matrix is orthogonal if and only if its columns and rows, considered as vectors, form an orthonormal set. The chapter presents a theorem that states that for every n × n real symmetric matrix A, there exists an n × n real orthogonal matrix P, such that P-1AP = D or, equivalently, such that PTAP = D, where D is a diagonal matrix. The chapter further presents Hermitian matrices. A Hermitian matrix is positive definite if and only if its eigenvalues are positive.