Recent advances in matrix theory

Book Reviews it as a reference to be kept in a convenient practice of the amperometric method, which place on the bookshelf. has certain decided advantages over many ENOSE. WITMER types of titrations and in various concentraUniversity of Hamburg tion ranges. As far as he is aware this reviewer Hamburg, Germany knows of no comparable work on the subject that has previously appeared. N. H. FURMAN RECENTADVANCES IN MATRIX THEORY,edPrinceton Untie&y ited by Hans Schneider. 142 pages, 6 X 9 Princeton, New Jersey in. Madison, Wiiconsin, University of Wisconsin Press, 1964. Price, $4.00. DIE MATHEMATISCHEN HILFSMII’ITEL DES The contents of this volume comprise six PHYSIKER~, by Erwin Madelung. 536 pages, lectures presented at a seminar held at the diagrams, 6 X 9 in. Text in German. Berlin, University of Wisconsin in October 1963. Springer-Verlag, 1964. Price, DM 49.70 In the article on “Characteristic Roots of (approx. $12.00). Non-negative Matrices,” Alfred Brauer surThe 6rst edition of Madelung’s Die Matheveys results related to the famous theorems mat&hen Hilfsmittel des Physikers appeared of Perron and Frobenius. Perron’s theorem in 1922. Mathematical physics has expanded states that if a matrix is positive, that is, if enormously since that time and the author has all its elements are positive, then it neceskept pace with the forward march of mathesarily has a greatest positive characteristic matical physics and has expanded his book root w. Furthermore, all other characteristic to include the newer subjects. roots lie in the circle [z 1$o. Brauer presents About two thirds of the text are devoted to a number of results, many attributed to him, the mathematical topics a physicist needs, and relating to the localization of the characteristic the last third is devoted to mathematical roots of arbitrary matrices. Subsequently he physics. Although a very useful book, it is applies these to generalized stochastic manot a textbook or treatise, but rather a brief trices. These are such that Z”k,laih =s for all summary of each topic covered. Thus differ- i, where the oik are the elements in the matrix. ential and integral calculus is covered in 22 Using these results he furnishes a new proof pages including a table of integrals coverfor Perron’s theorem, and for Frobenius’ ing two pages. Vector analysis is covered in Theorem. In particular he shows that the 37 pages and the special theory of relativity, greatest positive root of a positive matrix is in 12 pages. Nevertheless, the topics are rather simple and can be computed as exactly as thoroughly discussed because the various sec- needed. He shows that some of these results tions are more in the nature of a collection of can be generalized to power-positive matrices. important definitions, methods and formulas, These are such that all elements of the matrix but give no proofs. To a considerable extent A are real and some power of A is positive. the book is a collection of formulas with the The second article, by A. S. Householder, necessary explanations to describe their use. “Localization of the Characteristic Roots of It is thus a very useful reference for one Matrices,” discusses some inclusion and excluwho knows the subject under consideration sion theorems. These pertain to methods for from regular textbooks and courses. Such a the determination of regions in the complex person can look up a particular item quickly plane that necessarily do or do not contain without consulting a big textbook or treatise characteristic roots. The use of Rayleigh on that subject. The fact that the coverage quotients is discussed in localizing the characis quite extensive is shown in that the mathe- teristic roots of normal matrices. Another immatical part includes sections on group theory portant method centers around the idea of and the calculus of variations. In the part on defining norms of matrices using bounded mathematical physics there are sections on equilibrated convex bodies. By specializing to quantum theory, including quantum mechan- particular convex bodies many classical theoics and also thermodynamics. rems can be derived. The author merely menThe book contains an excellent index and tions, without further discussion, that these a list of references to the literature on each methods can be applied in studying the spectopic considered. This reviewer recommends tra of operators in Hilbert spaces.

Vol.

280,

No.

1, July

1965

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&d

Reviews

Marvin Marcus’article, “The Use of Multilinear Algebra for Proving Matrix Inequalities,” is rather difficult to summarize in a few words. Its primary concern is to provide estimates for the permanents of positive Hermitian and positive doubly stochastic matrices. The necessary background in Multilinear Algebra is provided so that the article is selfcontained. A. M. Ostrowski in “Positive Matrices and Functional Analysis” returns to Perron’s theorem. He takes a completely different approach from Brauer’s, and provides one proof attributed to him ss well as a discussion of results due to G. Birkhoff and also E. Hopf. This article is of particular significance to analysts since the connections between Perron’s theorem and Jentzsch’s theorem are pointed out. This discussion provides sharp bounds on the eigenvalues of such positive matrix and integral operators. Generalizations from the aforementioned operators to more general linear positive functionals are discussed in some detail. In “Matrices of Zeros and Ones in Combinatorial Mathematics” H. J. Ryser discusses some recent work in regard to a class of matrices known as tournament or round robin matrices. These arise in tournaments involving n players where each player plays every other player precisely once. They also find applications in statistical investigations involving paired comparisons, and in graph theory. A number of structure theorems are proved and discussed and the article concludes with some unsolved problems. The final article in this seminar is “On the Variation of the Characteristic Roots of a Finite Matrix Under Various Changes of its Elements” by Olga Taussky. The running theme is perturbation theory and a great many results attributed to a number of different workers are mentioned. Unfortunately no bibliography is included. Various statements are made regarding the changes in the characteristic roots of a matrix A if another matrix B is added to A, or if A is multiplied by B or some elements of A replaced by their absolute values or if some elements of A are permuted. This is a rather rich survey of results and it would be futile to try to survey it briefly. By and large the contents of these Proceedings are very useful and enlightening. Much work on matrices that is of current interest is

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summarized here and the basic results should be of value to mathematicians with diverse backgrounds. The authors discuss work relating to analysis, algebra, combinatorics and numerical analysis. The results found here will be of value to specialists, as well as to others interested in current work in matrix theory. HARRY HOCHSTADT

Departmentof Mathematics Polytechnic

Institute of Brooklyn Brooklyn, New York

PUMPS AND BLOWERS: TWO-PHASE FLOW, by

A. J. Stepanoff. 316 pages, diagrams, illustrations 6 X 9 in. New York, John Wiley and Sons, Inc., 1965. Price $12.50. Stepanoff’s new book supplements his widely used basic text by treating many specialized problems of interest to a professional engineer in the field. While the substance of the material has appeared in various technical papers the author has given, he has organized it into a single convenient text where the problems are presented in better perspective than was possible in the isolated papers. The book is divided into two sections. The 6rst deals with the application of cavitation data obtained with water to other liquids such as molten metals, hydrocarbons, and cryogenics. In addition to an extensive discussion of cavitation, the first section contains characteristic performance curves for pumps in which the impellers were mismatched relative to the pump casings. The second section deals with the general behavior of centrifugal pumps handling various mixtures such as solid-liquid, solid-gas, or gas-liquid. Correlations of the available test information obtained on two component systems are presented. A blending of theory and application throughout this book affords solutions to many problems that face the designer of centrifugal pumps and blowers. The treatment is eminently practical with empirical solutions suggested for numerous cases in which the existing theory does not provide an attractive approach. Engineers working in the field should find this volume a valuable and convenient aid. A. P. FRAAS AND M. E. LACKEY Oak Ridge Nat&al Laboratory Oak Ridge, Tennessee

Journal

of The Franklin

Institute