Matrices with totally positive powers and their generalizations

Abstract

Abstract. In this paper, eventually totally positive matrices (i.e. matrices all whose powers starting at some point are totally positive) are studied. We present a new approach to eventual total positivity which is based on the theory of eventually positive matrices. We mainly focus on the spectral properties of such matrices. We also study eventually J-sign-symmetric matrices and matrices, whose powers are P -matrices.Mathematics subject classification (2010): Primary 15A48; Secondary 15A18, 15A75.

On the Level-Slope-Curvature Effect in Yield Curves and Eventual Total Positivity

“…The crucial ingredient in order to prove these kinds of results is the theorem of Perron and Frobenius (see, for example, [18]). Interestingly, as is proved in [17] and [13] (see also Lemma 2.1 in [8]), a certain converse of the theorem of Perron and Frobenius is true. Nevertheless, the converse uncovers a condition that is stronger that the definition of LSC given by Lord and Pelsser, but, as we will argue later in this paper, is a condition that agrees more with the intuition behind LSC analysis.…”

“…Since the case GK 1 is totally characterized by eventually positive matrices, the open question is whether there is a similar characterization for GK k for any k. The answer is given in the next theorem, which is a generalization of the Theorem 7 in [13]. There are two reasons why we want to generalize the result in Theorem 7 in [13]. On one hand, as it was stated in the introduction, this result will provide an answer to the question posed by Lord and Pelsser; i.e., it will characterize the family of matrices satisfying LSC (with a new definition, which we will call strong LSC), and, on the other hand, it will strengthen the case made by Lekkos [14].…”

“…Since the case GK 1 is totally characterized by eventually positive matrices, the open question is whether there is a similar characterization for GK k for any k. The answer is given in the next theorem, which is a generalization of the Theorem 7 in [13]. There are two reasons why we want to generalize the result in Theorem 7 in [13].…”

“…(2) ρ bear any relationship with the eigenvectors and eigenvalues of the original matrix M ρ ? The answer to this question is at the heart of the following theorem due to Kronecker (see, for example [13]).…”

“…It turns out that looking at the number of zero-crossings of the eigenvectors is not enough; rather the condition on the number of zero-crossings has to be satisfied by all the vectors in certain subspaces. To show this we have relied heavily on some recent results in linear algebra, particularly the work of Kushel [13]. Understanding the structure of these matrices allows us to see why the spectral structure of covariance (or correlation) matrices is so similar in different interest rate markets.…”