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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]

In this paper, using Schur complements, we prove various inequalities in Euclidean Jordan algebras. Specifically, we study analogues of the inequalities of Fischer, Hadamard, Bergstrom, Oppenheim, and other inequalities related to determinants, eigenvalues, and Schur complements. [ABSTRACT FROM AUTHOR]