In this thesis we study a Geršgorin type theorem, spectral inequalities, and simultaneous stability of linear transformations in the setting of Euclidean Jordan algebras.

For complex square matrices, the Levy-Desplanques theorem asserts that a strictly diagonally dominant matrix is invertible. The well-known Geršgorin theorem on the location of eigenvalues is equivalent to this. In the first part of the thesis, we extend the Levy-Desplanques theorem to an object in a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. As a consequence, we prove a Geršgorin type theorem for the spectral eigenvalues of an object in a Euclidean Jordan algebra.

In matrix theory, the well known Cauchy's interlacing theorem states that if a row-column pair is deleted from a complex Hermitian matrix, then the eigenvalues of the resulting principal matrix interlace those of the original one. This result was generalized recently to the setting of simple Euclidean Jordan algebras in [13]. In the second part of the thesis, we present some consequences of this generalization and prove the dual form of the min-max theorem of Hirzebruch. One of the applications of the min-max theorem from matrix analysis is the problem of comparing the eigenvalues of Hermitian matrices A and B with those of A + B, proved by Weyl. Using the min-max theorem of Hirzebruch, we extend the Weyl's inequality to the setting of simple Euclidean Jordan algebras and study some consequences of it.

Using duality and complementarity ideas, and Z-transformations, in the third part of the thesis, we discuss ways of describing simultaneous stability of linear transformations on Hilbert spaces. As a consequence, we discuss the existence of common linear/quadratic Lyapunov functions for switched linear systems. In particular, we extend a recent result of Mason-Shorten on positive switched system with two constituent linear time-invariant systems to an arbitrary finite system.