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Let A and B be rectangular matrices. Then A is orthogonal to B if II A + f-LB II ?: II A II for every scalar f-L. Some approximation theory and convexity results on matrices are used to study orthogonality of matrices and answer an open problem of Bhatia and Semrl. © 2002 Elsevier Science Inc. All rights reserved.

The class of shape equivalences for a pair (C,K) of categories is the orthogonal of K, that is Σ = K. Then Σ is internally saturated (Σ = Σ). On the other hand, every internally saturated class of morphisms Σ ⊂ Mor(C), is the class of shape equivalences for some pair (C,K). Moreover, every class of shape equivalences Σ enjoys a calculus of left fractions and such a fact allows one to use techni...

Adámek and Sousa recently solved the problem of characterizing the subcategories K of a locally λ-presentable category C which are λ-orthogonal in C, using their concept of Kλ-pure morphism. We strenghten the latter definition, in order to obtain a characterization of the classes defined by orthogonality with respect to λ-presentable morphisms (where f :A B is called λ-presentable if it is a λ-...

The characterization of automorphisms of Bernstein algebras is an open problem. We only know some particular results. Previously we have characterized the automorphisms of quasiorthogonal, orthogonal, and strongly orthogonal weak Bernstein-Jordan algebras. In this paper we work on the minimal dimension with respect to quasiorthogonality, orthogonality, and strong orthogonality. We establish som...

In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formal...

A logic of orthogonality characterizes all “orthogonality consequences” of a given class Σ of morphisms, i.e. those morphisms s such that every object orthogonal to Σ is also orthogonal to s. A simple four-rule deduction system is formulated which is sound in every cocomplete category. In locally presentable categories we prove that the deduction system is also complete (a) for all classes Σ of...

The purpose of this paper is to introduce and discuss the concept of orthogonality in normed spaces. A concept of orthogonality on normed linear space was introduced. We obtain some subspaces of Banach spaces which are topologically complemented.