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A real n × n matrix M is said to be a P -matrix if all its principal minors are positive. In a recent paper Chua and Yi [2] described this property in terms of norm: There exists a γ > 0 such that for all nonnegative diagonal matrices D and vectors x , ‖ M x + D x ‖ ≥ γ ‖ x ‖ . In this paper, we introduce a block version of this property for a linear transformation defined on a product of normed or inner product spaces. In addition to relating this to (real) positive stability and positive principal minor properties, we study the invariance of this property by principal subtransformations and Schur complements. We also specialize this property to Z -transformations and to Euclidean Jordan algebras. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper [7], Gowda et al. extended Ostrowski–Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that whenever by the min–max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined bywith , , and denoting, respectively, the number of positive, negative, and zero eigenvalues, counting multiplicities. In this paper, we present a Peirce decomposition version of Wimmer’s result [13] and show that it is equivalent to the above result. In addition, we extend Higham and Cheng’s result ([8], Lemma 4.2) to the setting of Euclidean Jordan algebras. [Copyright &y& Elsevier]

Motivated by the similarities between the properties of Z-matrices on $$R^{n}_+$$ and Lyapunov and Stein transformations on the semidefinite cone $$\mathcal {S}^n_+$$ , we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Finally, we study the equivalence of Q and P properties of Z-transformations on symmetric cones. In particular, we prove such an equivalence on the Lorentz cone. [ABSTRACT FROM AUTHOR]

In matrix theory, Fu and Markham showed using majorization technique that if a Hermitian matrix satisfies certain conditions, then the matrix must be block-diagonal. In this paper, we extend this result to the setting of simple Euclidean Jordan algebras by using the Cauchy interlacing theorem and the Schur complement Cauchy interlacing theorem. [ABSTRACT FROM AUTHOR]