During this ebook the authors strengthen a thought of unfastened noncommutative services, in either algebraic and analytic settings. Such capabilities are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector house) to sq. matrices over one other module, which recognize the scale, direct sums, and similarities of matrices. Examples contain, yet will not be restricted to, noncommutative polynomials, strength sequence, and rational expressions. Motivation and notion for utilizing the speculation of unfastened noncommutative features frequently comes from loose likelihood. a huge software sector is "dimensionless" matrix inequalities; those come up, e.g., in a variety of optimization difficulties of approach engineering. between different similar parts are these of polynomial identities in jewelry, formal languages and finite automata, quasideterminants, noncommutative symmetric features, operator areas and operator algebras, and quantum regulate.

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