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Totally positive matrices. (English) Zbl 0613.15014

As a survey paper on totally positive matrices, this article enhances the earlier work of Gantmacher and Krein, and Karlin. There are seven short sections, each complete with definitions, theorems, proofs and references on: determinantal identities in light of tensor products and Schur complements; criteria for total positivity using sign-regularity; permanence of total positivity involving products and factorisations of totally positive matrices; oscillatory matrices; variation of signs; eigenvalues and eigenvectors and examples of totally positive matrices such as Hurwitz and certain Toeplitz matrices.
Reviewer: F.Uhlig

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A09 Theory of matrix inversion and generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
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