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High, low, and quantitative roads in linear algebra. (English) Zbl 0756.15002

In this survey of the present state and future prospects of “core” linear algebra, the author highlights the value of the different approaches (the use of elementary vs. sophisticated tools such as Lie algebra and algebraic geometry), and the use of computers to generate conjectures and prove theorems.
The topics considered are well described by the section headings: the numerical range; similarity invariants of principal submatrices; commutators; the triangle inequality; the facial structure of the unit ball; the Gerschgorin circle theorem; matrices, graphs, inertia, number theory; power embeddings and dilations; the Schubert calculus; the spectrum of the sum of Hermitian matrices; the Hadamard-Schur product; the exponential function; integral quadratic forms; the matrix valued numerical range; and inequalities with subtracted terms.
Each section deals with one or more problems, a description of the techniques used successfully to tackle them, an introduction to the literature, and questions which still remain. There is a bibliography of 145 items.
Reviewer: J.D.Dixon (Ottawa)

MSC:

15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
15-03 History of linear algebra
15-XX Linear and multilinear algebra; matrix theory

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Mathematica
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References:

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