5 - Eigenvalues and eigenvectors

Summary

In this chapter we review the spectral properties of totally positive matrices. A strictly totally positive matrix has positive, simple eigenvalues and the associated eigenvectors possess an intricate structure. Such is not the case for totally positive matrices. However, there is an intermediate set of matrices with the same spectral properties as strictly totally positive matrices. These matrices are called oscillation matrices. They shall be discussed in Section 5.1. In Section 5.2 we present the Gantmacher–Krein Theorem (Theorem 5.3) and give two quite different proofs thereof. This theorem contains the main spectral properties of oscillation matrices. In Section 5.3 we consider eigenvalues of the principal submatrices of such matrices and study their behaviour. We study in more detail the properties of eigenvectors of oscillation matrices in Section 5.4. Finally, in Section 5.5, we look at how the eigenvalues of oscillation matrices vary as functions of the elements of the matrix.

Oscillation matrices

Oscillation matrices are a class of matrices intermediary between totally positive and strictly totally positive matrices. They share the eigenvalue and eigenvector structure of strictly totally positive matrices.

Definition 5.1 An n × n matrix A is said to be an oscillation matrix if A is totally positive and some power of A is strictly totally positive.

Importantly, there are relatively simple criteria for determining if a totally positive matrix is an oscillation matrix.