2 - Criteria for total positivity and strict total positivity

Summary

In this chapter we discuss the problem of establishing determinantal criteria for when a matrix is totally positive or strictly totally positive. Totally different criteria will be discussed in the next chapter on variation diminishing.

In Section 2.1 we prove Fekete's Lemma and some of its consequences. The most notable thereof is Theorem 2.3, which states that for a matrix to be strictly totally positive it suffices to prove that all its minors composed of the first k rows and k consecutive columns and all its minors composed of the first k columns and k consecutive rows are strictly positive for all possible k. We apply the results of Section 2.1 in Section 2.2, where we prove that strictly totally positive matrices are dense in the class of totally positive matrices, and provide proofs of Propositions 1.9 and 1.10 from Chapter 1.

In Section 2.3 we discuss triangular matrices, detail determinantal criteria for when such matrices are totally positive, and in Section 2.4 we consider the LDU-factorization of strictly totally positive and totally positive matrices. We will return to the study of factorizations of totally positive matrices in Chapter 6. In Section 2.5 we consider determinantal criteria for when a matrix is totally positive. The results are nowhere near as elegant as those valid for strictly totally positive matrices.

In Section 2.6 we prove a recent surprising and beautiful result of O. M. Katkova and A. M. Vishnyakova (completing work initiated by T. Craven and G. Csordas).