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A determinantal inequality for positive semidefinite matrices
No full textLet A, B, C be n à n positive semidefinite matrices. It is known that det(A + B + C) + det C ⥠det(A + C) + det(B + C), which includes det(A + B) ⥠det A + det B as a special case. In this article, a relation between these two inequalities is proved, namely, det(A + B + C) + det C â (det(A + C) + det(B + C)) ⥠det(A + B) â (det A + det B)
On the maximal angle between copositive matrices
No full textHiriart-Urruty and Seeger have posed the problem of finding the maximal possible angle θ_{max}(C_n) between two copositive matrices of order n [J.-B. Hiriart-Urruty and A. Seeger. A variational approach to copositive matrices. SIAM Rev., 52:593â629, 2010.]. They have proved that θ_{max}(C_2) = (3/4)pi and conjectured that θ_{max}(C_n) is equal to (3/4)pi for all n ⥠2. In this note, their conjecture is disproven by showing that lim_{nââ} θ_{max}(C_n) = pi. The proof uses a construction from algebraic graph theory. The related problem of finding the maximal angle between a nonnegative matrix and a positive semidefinite matrix of the same order is considered in this paper
Inequalities for relative operator entropies
No full textIn this paper, operator inequalities are provided for operator entropies transformed by a strictly positive linear map. Some results by Furuichi et al. [S. Furuichi, K. Yanagi, and K. Kuriyama. A note on operator inequalities of Tsallis relative operator entropy. Linear Algebra Appl., 407:19â31, 2005.], Furuta [T. Furuta. Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications. Linear Algebra Appl., 412:526â537, 2006.], and Zou [L. Zou. Operator inequalities associated with Tsallis relative operator entropy. Math. Inequal. Appl., 18:401â406, 2015.] are extended. In particular, the obtained inequalities are specified for relative operator entropy and Tsallis relative operator entropy. In addition, some bounds for generalized relative operator entropy are established
Schur ideals and homomorphisms of the semidefinite cone
No full textThe semidefinite cone Kn consisting of all n by n real symmetric positive semidefinite matrices is considered. A set I in Kn is said to be a Schur ideal if it is closed under addition, multiplication by nonnegative scalars, and Schur multiplication by any element of Kn. A Schur homomorphism of Kn is a mapping of Kn to itself that preserves addition, (nonnegative) scalar multiplication and Schur products. This paper is concerned with Schur ideals and homomorphisms of Kn. It shows that in the topology induced by the trace inner product, Schur ideals in Kn need not be closed, all finitely generated Schur ideals are closed, and in K2, a Schur ideal is closed if and only if it is a principal ideal. It also characterizes Schur homomorphisms of Kn and, in particular, shows that any Schur automorphism of Kn is of the form Φ(X) = P XPT for some permutation matrix P
On the max-algebraic core of a nonnegative matrix
No full textThe max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. This paper investigates the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrice
Change of the *-congruence canonical form of 2-by-2 matrices under perturbations
No full textIt is constructed the Hasse diagram for the closure ordering on the sets of *congruence classes of 2 × 2 matrices. In other words, it is constructed the directed graph whose vertices are 2 × 2 canonical complex matrices for *congruence and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is *congruent to B
Water Resources Internship Report: Grand Teton National Park
No full textThe objective of this project was to consolidate all previous work on water rights in the park into a single geodatabase that could be updated and built on in the future. Priority areas specifically for this project were Cottonwood and Spread Creeks, with the goal being to identify all water rights and associated ditches being diverted off of these creeks
