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 object and its principal components and extend some well known matrix theory theorems and inequalities to the setting of simple Euclidean Jordan algebras.
(February 29, 2008, Revised July 24, 2009)