The Cauchy interlacing theorem in simple Euclidean Jordan algebras and some consequences

Abstract

In this article, based on the min-max theorem of Hirzebruch, we formulate and prove the Cauchy interlacing theorem in simple Euclidean Jordan algebras. As a consequence , we relate the inertias of an element and its principal components and extend some well known matrix theory theorems and inequalities to the setting of simple Euclidean Jordan algebras.

Some majorization inequalities in Euclidean Jordan algebras

“…Hence, the first inequality in (11) becomes equality by Corollary 4.10 in [5] and the last inequality in (11) becomes equality as λ ↓ i (x) = 1 a p/q tr,p (λ ↓ i (a)) p−1 . Hence, a, x = a tr,p .…”

“…Then two inequalities in (11) become equalities. Thus, a and x operator commute by Corollary 4.10 in [5] as the first equality in (11) and λ ↓ i (x) = c(λ ↓ i (a)) p−1 for some positive number c as the second equality in (11). Since a and x operator commute, there exists a Jordan frame {e 1 , e 2 , .…”

Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras

“…It has been observed in [10] that in a simple algebra of rank r , the following Minkowski type inequality holds for all x, y ≥ 0:…”

More results on Schur complements in Euclidean Jordan algebras

“…where a := An analogous result holds in any Euclidean Jordan algebra [10]: If x ≥ 0 in J with Peirce decomposition x = i≤j x ij with respect to any Jordan frame {e 1 , e 2 , . .…”

“…. , r. Since these are the "leading principal minors" of x, by Problem 5, VI.4 in [6] or Corollary 4.5 in [10], x > 0. ⊓ ⊔…”