Schur complements, Schur determinantal and Haynsworth inertia formulas in Euclidean Jordan algebras

In this article, we study the concept of Schur complement in the setting of Euclidean Jordan algebras and describe Schur determinantal and Haynsworth inertia formulas.     

More results on Schur complements in Euclidean Jordan algebras

In a recent article Gowda and Sznajder (Linear Algebra Appl 432:1553–1559, 2010) studied the concept of Schur complement in Euclidean Jordan algebras and described Schur determinantal and Haynsworth inertia formulas. In this article, we establish some more results on the Schur complement. Specifically, we prove, in the setting of Euclidean Jordan algebras, an analogue of the Crabtree-Haynsworth quotient formula and show that any Schur complement of a strictly diagonally dominant element is strictly diagonally dominant. We also introduce the concept of Schur product of a real symmetric matrix and an element of a Euclidean Jordan algebra when its Peirce decomposition with respect to a Jordan frame is given. An Oppenheim type inequality is proved in this setting.     

A note on an inequality involving Jordan product in Euclidean Jordan algebras

In a most recent paper, Yang et al. (Appl Math Comput 230:616–628, 2014) proved an important inequality on a simple Euclidean Jordan algebra by using a case-by-case analysis. In this paper, we improve this inequality in any Euclidean Jordan algebras and the proof is not based on a case-by-case analysis.     

Some inequalities involving eigenvalues of the Neumann Laplacian

This paper is concerned with the eigenvalues of the Neumann Laplacian on various classes of domains of given measure: simply‐connected Lipschitz planar domains, n‐sided planar polygons and smooth N‐dimensional domains. In each case, we consider some quantities involving low eigenvalues of the Neumann Laplacian for which we obtain new inequalities. Moreover, we sharpen a universal bound derived by M. Ashbaugh and R. Benguria for sum of reciprocal of Neumann eigenvalues. Our investigations make use of some properties of conformal mappings, Bessel functions, symmetric domains or some isoperimetric inequalities for moments of inertia. Copyright © 2013 John Wiley & Sons, Ltd.     

On perturbation bounds of eigenvalues in Euclidean Jordan Algebras

In matrix theory, C.-K. Li and R.-C. Li gave the best bound of the difference between eigenvalues of a real/complex Hermitian matrix and the matrix after removing off-diagonal blocks. In this paper, we extend this result to the setting of simple Euclidean Jordan Algebras.     

Strict double diagonal dominance in Euclidean Jordan algebras

In the first part of the paper, we deal with Euclidean Jordan algebraic generalizations of some results of Brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs. As a consequence, the theorems of Brauer–Ostrowski and Brauer on the location of eigenvalues are extended to the setting of Euclidean Jordan algebras. In the second part, motivated by the work of Li and Tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses, we present some inter-relations between the H-property, generalized strict diagonal dominance, invertibility, and strict double diagonal dominance in Euclidean Jordan algebras. In addition, we show that in a Euclidean Jordan algebra, the Schur complements of a strictly doubly diagonally dominant element inherit this property.     

Some properties of Schur complements and diagonal-Schur complements of diagonally dominant matrices

In this paper, we prove that the diagonal-Schur complement of a strictly doubly diagonally dominant matrix is strictly doubly diagonally dominant matrix. The same holds for the diagonal-Schur complement of a strictly generalized doubly diagonally dominant matrix and a nonsingular H-matrix. We point out that under certain assumptions, the diagonal-Schur complement of a strictly doubly (doubly product) γ-diagonally dominant matrix is also strictly doubly (doubly product) γ-diagonally dominant. Further, we provide the distribution of the real parts of eigenvalues of a diagonal-Schur complement of H-matrix. We also show that the Schur complement of a γ-diagonally dominant matrix is not always γ-diagonally dominant by a numerical example, and then obtain a sufficient condition to ensure that the Schur complement of a γ-diagonally dominant matrix is γ-diagonally dominant.     

Sufficiency of linear transformations on Euclidean Jordan algebras

We extend the row-sufficiency and column-sufficiency of a linear transformation from to the setting of Euclidean Jordan algebras. For linear complementarity problems over symmetric cones, we show that the column-sufficiency along with Cross Commutative property is equivalent to the convexity of the solution set, while the row-sufficiency is necessary for the existence of solutions under some conditions. The work was partly supported by the National Natural Science Foundation of China (10671010, 70640420143).     

Schur complements and statistics

In this paper we discuss various properties of matrices of the type

$\text{S}\text{=}\text{H}\text{?}\text{GE}{}^{\mathrm{?1}}\text{F}$

, which we call the Schur complement of E in

$\text{A}\text{=}\text{E}\text{F}\text{G}\text{H}$

The matrix E is assumed to be nonsingular. When E is singular or rectangular we consider the generalized Schur complements $\text{S}\text{=}\text{H}\text{?}\text{GE}{}^{\mathrm{?}}\text{F}$, where E^? is a generalized inverse of E. A comprehensive account of results pertaining to the determinant, the rank, the inverse and generalized inverses of partitioned matrices, and the inertia of a matrix is given both for Schur complements and for generalized Schur complements. We survey the known results in a historical perspective and obtain several extensions. Numerous applications in numerical analysis and statistics are included. The paper ends with an exhaustive bibliography of books and articles related to Schur complements.     

Some probability inequalities connected with Schur functions

Bounds for several integrals (tail probabilities, for example) are established by showing that each integral is a Schur function.     

Jordan quadratic SSM-property and its relation to copositive linear transformations on Euclidean Jordan algebras

In this paper, we introduce Jordan quadratic SSM-property and study its relation to copositive linear transformations on Euclidean Jordan algebras. In particular, we study this relationship for normal Z-transformations, Lyapunov-like transformations and cone invariant transformations.     

Maximal Jordan algebras of matrices with bounded number of eigenvalues

We consider maximal Jordan algebras of matrices with bounded number of eigenvalues. Up to simultaneous similarity we list all irreducible algebras of that kind, and we also give a list of some reducible such algebras. We also study automorphisms of Jordan algebras of matrices. Research supported in part by the NSERC of Canada and by the Ministry of Science and Technology of Slovenia.     

Schur algebras of Brauer algebras,II

A classical problem of invariant theory and of Lie theory is to determine endomorphism rings of representations of classical groups, for instance of tensor powers of the natural module (Schur–Weyl duality) or of full direct sums of tensor products of exterior powers (Ringel duality). In this article, the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined. These are shown to be isomorphic to Schur algebras of Brauer algebras as defined in Henke and Koenig (Math Z 272(3–4):729–759, 2012). This implies structural properties of the endomorphism rings, such as double centraliser properties, quasi-hereditary, and a universal property, as well as a classification of simple modules.     

A note on the Ostrowski-Schneider type inertia theorem in Euclidean Jordan algebras

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 In(a)=In(x) whenever a°x>0 by the min-max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined by

In(x):=(π(x),ν(x),δ(x)),     

Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field . We show that the natural homomorphism from the Brauer algebra to the endomorphism algebra of the tensor space as a module over the symplectic similitude group (or equivalently, as a module over the symplectic group ) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for to the endomorphism algebra of as a module over , is derived as an easy consequence of S. Oehms's results [S. Oehms, J. Algebra (1) 244 (2001), 19-44].