Critical exponents: old and new

Abstract

Let P be a class of matrices, and let A be an m-by-n matrix in the class; consider some continuous powering, A {t} . The critical exponent of P, if it exists, with respect to the powering is the lowest power g(P) such that for any matrix B ∈ P , B {t} ∈ P ∀ t > g(P). For powering relative to matrix multiplication in the traditional sense, hereafter referred to as conventional multiplication, this means that A t is in the specified class for all t > g C (P). For Hadamard multiplication, similarly, A (t) is in the class ∀ t > g H (P). This paper considers two questions for several classes P (including doubly nonnegative and totally positive): 1) does a critical exponent g(P) exist? and 2) if so, what is it? For those where no exact result has been determined, lower and upper bounds are provided.

Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

“…Finally, some of our counterexamples also settle the Hadamard critical-exponent problem [23] for TN or TP matrices that are general, symmetric or Hankel.…”

“…Less trivially, the TN 3 /TP 3 case is handled by the following result [23,Theorem 4.2] (see also [9, pp. 179-180]):…”

“…Example 5.5 shows that Proposition 5.6 cannot be extended to any t ∈ (0, 1)∪(1, 2), even if the matrix is Hankel and TP. It follows, in the language of [23], that the Hadamard critical exponent for 4-by-4 symmetric TN or TP matrices is 2.…”

Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity

“…Once again, we are able to obtain a complete classification of all real powers preserving the five aforementioned Loewner properties. Similar to many settings in the literature (see [18]), one can define Hadamard critical exponents for positivity, monotonicity, convexity, and super-additivity for P k n -these are the phase transition points akin to [8]. From Theorem 1.2, we immediately obtain the Hadamard critical exponents (CE) for the four Loewner properties for matrices with rank constraints: Corollary 1.4.…”

“…Indeed, the study of critical exponents -and more generally of functions preserving a form of positivity -is an interesting and important endeavor in a wide variety of situations, and has been studied in many settings (see e.g. [21,17,10,22,18]).…”