218 - Österreichische Mathematische Gesellschaft

A. Pinkus: Totally Positive Matrices. (Cambridge Tracts Mathematics 181.)
Cambridge University Press, 2010, xi+182 S. ISBN 978-0-521-19408-2 H/b
£ 40,–.
The book under review is the first one dealing exclusively with the theory of totally
positive matrices. An m×n matrix is called totally positive (TP) resp. strictly
totally positive (STP) if all its minors are nonnegative resp. positive. An oscillation
matrix A is a TP matrix such that some power A p is STP. The theory developed
from various problems in (applied) analysis. First contributions are by
I. J. Schoenberg from the early 1930s who estimated the number of real zeros of
a polynomial. The major classical texts blending analysis and matrix theory are
Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems by
F. R. Gantmacher and M. G. Krein (1950) and Total Positivity I by A. S. Karlin
(1968). A first comprehensive account of the state of the art was given in a survey
article by T. Ando (1987). The main intention of this monograph by Pinkus is to
cover the research progress done since the appearance of that paper and to give a
self-contained presentation of the whole topic.
There are six chapters. Chapter 1 presents basic properties of TP and STP matrices,
discusses nonsingularity and rank, and several determinantal inequalities
(e.g., generalized Hadamard inequalities). Chapter 2 deals with criteria for the
(scrict) total positivity (including Fekete’s lemma), density properties of STP matrices,
triangular total positivity, and LDU factorizations. Chapter 3 gives an introduction
to variation diminishing properties of TP matrices (introduced by Schoenberg).
Chapter 4 presents various examples of TP (STP) matrices, particularly a
series of subclasses that are closed with respect to the Hadamard product (Cauchy,
Green, Jacobi, Hankel, Toeplitz, Hurwitz matrices). Eigenvalues and eigenvectors
are dealt with in Chapter 5. There, the Gantmacher-Krein theorem on oscillation
matrices is proved and eigenvalues of principal submatrices (including interlacing
properties) are investigated. Chapter 6 is devoted to factorizations of TP (STP)
matrices.
At the end of each chapter there is a remark section presenting numerous useful
historical remarks to the genesis of the topic and references to the literature. An
afterword gives tribute to the founders of the theory, Schoenberg, Krein, Gantmacher,
and Karlin. The monograph closes with an extensive bibliography and
with short author and subject indexes, as well.
The exposition is very clear and a pleasure to read. All the proofs are given in full
details. As the author points out explicitly, he deliberately focused on the theory
only. So, one or the other reader might regret the lack of applications in this book.
Yet Totally Positive Matrices serves as an excellent source to get acquainted with
the theory of total positivity and can be recommended to any matrix theorist but
also to analysts working in the field of total positivity.
[Addendum: An even more comprehensive monograph on the same topic, Totally
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