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.

Keywords:

• Euclidean Jordan algebras
• quadratic representations
• min–max theorem of Hirzebruch
• Cauchy interlacing theorem
• Schur's theorem
• Hadamard's inequality
• Fan's trace inequality

AMS Subject Classifications::

• 15A33
• 17C20
• 17C55

Acknowledgements

We wish to thank the referee for pointing out reference 20 and raising (converse) questions which resulted in Section 5.