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Functional models in linear algebra. (English) Zbl 0756.15001

This is an expository paper surveying linear algebra from the point of view of functional models. Functional models — largely developed by the author — give a basis free approach to linear algebra with a level of abstraction intermediate between abstract module theory and matrix theory.
The flavour of the theory is given by Theorem 3.2: Let \(A\) be a linear operator on the \(m\)-dimensional vector space \(F^ m\) over a field \(F\), let \(F^ m[x]\) be the space of “polynomials” in \(x\) with coefficients from \(F^ m\), and let \(S_ +\) denote the shift operator (multiplication by \(x)\). Then \(A\) is isomorphic to \(S_ +\) acting on some explicitly defined quotient module of \(F^ m[x]\).
Thus one may hope to understand the operators on \(F^ m\) by studying the corresponding quotient modules of \(F^ m[x]\) under \(S_ +\). The theme of this long paper is that functional models give a powerful method which can simplify proofs and illuminate connections between linear algebra and opertor theory, approximation theory, moment problems, interpolation and control theory. The author includes examples and a historical context. The bibliography includes over 100 items.
Reviewer: J.D.Dixon (Ottawa)

MSC:

15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
16D10 General module theory in associative algebras
15-03 History of linear algebra
00A15 Bibliographies for mathematics in general
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