Henryk Minc

A survey of matrix theory and matrix inequalities

Written for advanced undergraduate students, this highly regarded book presents an enormous amount of information in a concise and accessible format. Beginning with the assumption that the reader has never seen a matrix...

Upper bounds for permanents of

This invention relates to a printing device of the type having a carriage which supports a printing roll. In order to perform the printing operation, the carriage is pivotally and slidably mounted on a slide rod. An arm of the carriage is mounted in a housing and is provided with a guide roller that is located in the same vertical plane as the printing roll. The guide roller cooperates with a guide means that extends parallel to the slide rod to keep the printing roll in engagement with a printing anvil and to enable the printing roll to swing upwardly into a raised position when arriving at one of the ends of the slide rod.

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This is a sequel to the authors monograph Permanents a survey of developments in the theory of permanents in the years which followed the publication of the monograph. In particular, it contains Egoryĉevs proof of the van der Waerden conjecture, and a description of the current status of other conjectures. A comprehensive list of papers and books, either published during the period 1978–1981 or awaiting publication, is appended, together with an addenda to the bibliography in the monograph for the years preceding 1978.

-matrices

In a recent investigation of a conjecture on an upper bound for permanents of (0, 1)-matrices ( 2 ) we obtained some inequalities involving the function ( r !) 1/ r which are of interest in themselves. Probably the most interesting of them, and certainly the hardest to prove, is the inequality where o( r = ( r !) 1/ r . In the present paper we prove (1) and other inequalities involving the function o( r .

Theory of permanents 1978–1981

A study of properties of matrices with minimum permanent in a face of the polyhedron of doubly stochastic n × n matrices. The minima are determined for certain faces.

Some Inequalities involving ( r !) 1/ r

It is shown that there exists an /i-square matrix all whose eigenvalues and n-1 of whose entries are arbitrarily prescribed. This result generalizes a theorem of L. Mirsky. It is also shown that there exists an «-square matrix with some of its entries prescribed and with simple eigenvalues, provided that n of the nonprescribed entries lie on a diagonal or, alternatively, provided that the number of prescribed entries does not exceed In—2. A well-known result of L. Mirsky [3] states essentially that, given any 2n—1 complex numbers Ax, ■ • • , Xn, ax, • • • , an_x, there exists an nsquare matrix with eigenvalues A,, • • • , Xn and n—\ of its main diagonal entries equal to ax, • • ■ , an_x. Related results for matrices over general fields were also obtained by Farahat and Ledermann [1]. We show that the restriction of the n—\ prescribed entries to the main diagonal is unnecessary. We first investigate the conditions under which there exists a matrix with prescribed entries and simple eigenvalues. A position in a matrix in which some entries have been prescribed is said to be free, if there is no prescribed entry in that position. By a diagonal in an «-square matrix we mean a set of n positions no two of which are in the same row or in the same column; i.e., positions (i, o(i)), /=1, •• • , n, for some permutation a. Theorem 1. Let ax, • • • , ani_n be n2—nprescribed complex numbers and let (it,ft), /= 1, ■ • • , n2—n, be prescribed different positions in an n-square matrix, such that the n remaining free positions form a diagonal of the matrix. Then there exists an n-square matrix with simple eigenvalues and with the prescribed entries at in the prescribed positions (it,jt), t= 1, • • • , n2—n. The number n of free positions cannot, in general, be decreased. Received by the editors June 1, 1971. AMS 1969 subject classifications. Primary 1525.

Minimum permanents of doubly stochastic matrices with prescribed zero entries

A theorem of Marcus and Moyls on linear transformations on matrices preserving rank 1 and a classical result of Frobenius on determinant preservers are re-proved by elementary matrix methods.

Eigenvalues of matrices with prescribed entries

Abstract It was shown by the author in a recent paper that a recurrence relation for permanents of (0, 1)-circulants can be generated from the product of the characteristic polynomials of permanental compounds of the companion matrix of a polynomial associated with (0, 1)-circulants of the given type. In the present paper general properties of permanental compounds of companion matrices are studied, and in particular of convertible companion matrices, i.e., matrices whose permanental compounds are equal to the determinantal compounds after changing the signs of some of their entries. These results are used to obtain formulas for the limit of the n th root of the permanent of the n × n (0, 1)-circulant of a given type, as n tends to infinity. The root-squaring method is then used to evaluate this limit for a wide range of circulant types whose associated polynomials have convertible companion matrices.

Linear transformations on matrices: rank 1 preservers and determinant preservers

Abstract Let A be an n-square (0, 1)-matrix, let ri denote the i-th row sum of A, i=1, …, n, and let per (A) denote the permanent of A. Then per ( A ) ≤ ∏ i = 1 n r i + 2 1 + 2 where equality can occur if and only if there exist permutation matrices P and Q such that PAQ is a direct sum of 1-square and 2-square matrices all of whose entries are 1.

Permanental compounds and permanents of (0, 1)-circulants

Abstract A general method is developed for constructing linear recurrence formulas expressing the permanents of (0, 1)-circulants in terms of permanents of (0, 1)-circulants of the same type of lower orders. The method extends to all (0, 1)-circulants a result of N. Metropolis, M. L. Stein, and P. R. Stein on permanents of (0, 1)-circulants of a special type.