Simultaneous kernels of matrix Hadamard powers

Abstract

In previous work [Adv. Math. 298:325-368, 2016], the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix were described. Key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well-known theorem of Hershkowitz, Neumann, and Schneider, which classifies the Hermitian positive semidefinite matrices whose entries are 0 or 1 in modulus. In this paper, we show that each of these results extends to a larger class of matrices which we term 3-PMP (principal minor positive).Date: March 26, 2018. 2010 Mathematics Subject Classification. 15B48 (primary); 15A21 (secondary).

Matrix compression along isogenic blocks

“…As a result, the above identity does not admit a uniform generalization, so we cannot naively adapt the previous proof to show that the subspace in (2) contains that in (1). There is a similar issue when α is not greater than n N , which cannot be resolved by modifying the arguments in [5].…”

“…However, for a general matrix A ∈ C N ×N , this extra property need not hold for π G (A), unless either G = {1}, or A ∈ P N (C) and G ⊂ S 1 . In fact, as shown in [5,Proposition 4.6], in the latter case the requirement in Theorem 2.1 that A is positive semidefinite may be relaxed by requiring A to be 3-PMP : every principal minor of size no more than 3 × 3 is non-negative.…”

“…Definition 3.1. ( [5]) Given an integer k ≥ 0, a complex Hermitian matrix is said to be k-PMP (principal minor positive) if every l × l principal minor is non-negative, for all l ∈ {1, . .…”

“…Remark 3.4. One subtlety in adapting the proof in [5] of Theorem 3.2 to these variants is that a key matrix identity relating…”

Hadamard Powers and Kernel Perceptrons

“…A prominent result is Schur's product theorem [9], which states that the Hadamard product preserves definiteness. Definiteness, rank, and other properties of Hadamard powers have been discussed in [10,11,12,13,14,15,1,16,17]. Applications of Hadamard products in artificial intelligence can be found in [18].…”