Transportation matrices with staircase patterns and majorization

We consider some questions concerning transportation matrices with a certain nonzero pattern. For a given staircase pattern we characterize those row sum vectors R and column sum vectors S such that the corresponding class of transportation matrices with the given row and column sums and the given pattern is nonempty. Two versions of this problem are considered. Algorithms for finding matrices in these matrix classes are introduced and, finally, a connection to the notion of majorization is discussed.     

A group majorization ordering for Euclidean distance matrices

This paper investigates some properties of Euclidean distance matrices (EDMs) with focus on their ordering structure. The ordering treated here is the group majorization ordering induced by the group of permutation matrices. By using this notion, we establish two monotonicity results for EDMs: (i) The radius of a spherical Euclidean distance matrix (spherical EDM) is increasing with respect to the group majorization ordering. (ii) The larger an EDM is in terms of the group majorization ordering, the more spread out its eigenvalues are. Minimal elements with respect to this ordering are also described.     

Row-ordering schemes for sparse givens transformations. II. implicit graph model

A new graph model is presented to study the row annihilation and row ordering problems in the QR decomposition of sparse matrices using Givens rotations. The graph-theoretic results obtained can be used to derive good row orderings for certain column orderings, such as width-1 and width-2 nested dissection orderings. This model is different from the bipartite-graph model introduced in [6]. We refer to the new model as implicit because the rows are not represented explicitly by nodes, in contrast to the bipartite-graph model, where the rows are represented by nodes in a bipartite graph.     

Properties of (0, 1)-matrices without certain configurations

We generalize results of Ryser on (0, 1)-matrices without triangles, 3 × 3 submatrices with row and column sums 2. The extremal case of matrices without triangles was previously studied by the author. Let the row intersection of row i and row j (i ≠ j) of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do not 0 otherwise. For matrices satisfying some conditions on forbidden configurations and column sums ? 2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. The extremal matrices with m rows and $\text{(}\text{m}\text{2}\text{)}$ distinct columns have a unique SDR of pairs of rows with 1's. A triangle bordered with a column of 0's and its (0, 1)-complement are also considered as forbidden configurations. Similar results are obtained and the extremal matrices are closely related to the extremal matrices without triangles.     

Convolution operators on Lebesgue spaces of an n-dimensional cube

This paper is concerned with the problem of diagonally scaling a given nonnegative matrix a to one with prescribed row and column sums. The approach is to represent one of the two scaling matrices as the solution of a variational problem. This leads in a natural way to necessary and sufficient conditions on the zero pattern of a so that such a scaling exists. In addition the convergence of the successive prescribed row and column sum normalizations is established. Certain invariants under diagonal scaling are used to actually compute the desired scaled matrix, and examples are provided. Finally, at the end of the paper, a discussion of infinite systems is presented.     

Properties of (0, 1)-matrices with no triangles

We study (0, 1)-matrices which contain no triangles (submatrices of order 3 with row and column sums 2) previously studied by Ryser. Let the row intersection of row i and row j of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do and a zero otherwise. For matrices with no triangles, columns sums ?2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. We then study the extremal (0, 1)-matrices with no triangles, column sums ?2, distinct columns, i.e., those of size mx(^m₂). The number of columns of column sum l is m ? l + 1 and they form a (l ? 1)-tree. The ((^m₂)) columns have a unique SDR of pairs of rows with 1's. Also, these matrices have a fascinating inductive buildup. We finish with an algorithm for constructing these matrices.     

Generalized fuzzy matrices

We give a systematic development of fuzzy matrix theory. Many of our results generalize to matrices over the two element Boolean algebra, over the nonnegative real numbers, over the nonnegative integers, and over the semirings, and we present these generalizations. Our first main result is that while spaces of fuzzy vectors do not have a unique basis in general they have a unique standard basis, and the cardinality of any two bases are equal. Thus concepts of row and column basis, row and column rank can be defined for fuzzy matrices. Then we study Green's equivalence classes of fuzzy matrices. New we give criteria for a fuzzy matrix to be regular and prove that the row and column rank of any regular fuzzy matrix are equal. Various inverses are also studied. In the next section, we obtain bounds for the index and period of a fuzzy matrix.     

Generalized doubly stochastic and permutation matrices over a ring

Let R be a ring with unity. A combinatorial argument is used to show that the R-module Δ_n(R) of all n × n matrices over R with constant row and column sums has a basis consisting of permutation matrices. This is used to characterize orthogonal matrices which are linear combinations of permutation matrices. It is shown that all bases of Δ_n(R) consisting of permutation matrices have the same cardinality, and other properties of bases of Δ_n(R) are investigated.     

Linear relations of refined enumerations of alternating sign matrices

In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of n×n alternating sign matrices where the top row as well as the left and the right column is fixed and the number of n×n alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely.     

Biproportional scaling of matrices and the iterative proportional fitting procedure

A short proof is given of the necessary and sufficient conditions for the convergence of the Iterative Proportional Fitting procedure. The input consists of a nonnegative matrix and of positive target marginals for row sums and for column sums. The output is a sequence of scaled matrices to approximate the biproportional fit, that is, the scaling of the input matrix by means of row and column divisors in order to fit row and column sums to target marginals. Generally it is shown that certain structural properties of a biproportional scaling do not depend on the particular sequence used to approximate it. Specifically, the sequence that emerges from the Iterative Proportional Fitting procedure is analyzed by means of the L ₁-error that measures how current row and column sums compare to their target marginals. As a new result a formula for the limiting L ₁-error is obtained. The formula is in terms of partial sums of the target marginals, and easily yields the other well-known convergence characterizations.     

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.     

Intrinsic products and factorizations of matrices

We say that the product of a row vector and a column vector is intrinsic if there is at most one nonzero product of corresponding coordinates. Analogously we speak about intrinsic product of two or more matrices, as well as about intrinsic factorizations of matrices. Since all entries of the intrinsic product are products of entries of the multiplied matrices, there is no addition. We present several examples, together with important applications. These applications include companion matrices and sign-nonsingular matrices.     

Some properties of the solution space of the N-city traveling-salesman problem

For the matrix of variables in the N-city traveling-salesman problem, consider both the N row and the N column vectors. An orthogonality condition involving products of row and column vectors is shown to eliminate subtours. Also, a group representation of the problem is given to observe properties of the solution space. The matrix of variables is subsequently decomposed into a product of elementary transposition matrices. Numerous examples are provided to illustrate the properties of the problem.     

Interlacing and degree conditios for invariat polynomials

It is well known that the invariant factors of a polynomial matrix interlace with ihose of one of its submatrices. If we restrict the degrees of the entries of our matrices in some prescribed way then the invariant factors of such matrices and submatrices satisfy other conditions that may be predicted and, in some cases, even completely described. That is the kind of problem we deal with Our main result is a sort of interlacing theorem for row proper matrices. Convexity and majorization also enter into play.     

Representing matrices

Let A be a nonnegative real matrix whose column set is countable. We give a necessary and sufficient condition on A for the existence of a nonnegative matrix B, B ? A, with column sums equal to prescribed numbers, and row sums not greater than prescribed numbers. This is a generalization of a result of Damerell and Milner, who solved the problem for (0, 1) matrices.     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

A sign test for detecting the equivalence of Sylvester Hadamard matrices

In this paper we show that every matrix in the class of Sylvester Hadamard matrices of order 2 ^k under H-equivalence can have full row and column sign spectrum, meaning that tabulating the numbers of sign interchanges along any row (or column) gives all integers 0,1,...,2 ^k  − 1 in some order. The construction and properties of Yates Hadamard matrices are presented and is established their equivalence with the Sylvester Hadamard matrices of the same order. Finally, is proved that every normalized Hadamard matrix has full column or row sign spectrum if and only if is H-equivalent to a Sylvester Hadamard matrix. This provides us with an efficient criterion identifying the equivalence of Sylvester Hadamard matrices.     

Non-singular 0-1 matrices with constant row and column sums

This paper explicitly constructs non-singular 0-1 matrices of dimensions n with constant row and column sum k, subject only to the conditions 0<k<n and (k,n)≠(2,4). An application of such matrices to finite geometries can be found in [1].     

Matrix enlarging methods and their application

This paper explores several methods for matrix enlarging, where an enlarged matrixà is constructed from a given matrixA. The methods explored include matrix primitization, stretching and node splitting. Graph interpretations of these methods are provided. Solving linear problems using enlarged matrices yields the answer to the originalAx=b problem.à can exhibit several desirable properties. For example,à can be constructed so that the valence of any row and/or column is smaller than some desired number (≥4). This is beneficial for algorithms that depend on the square of the number of entries of a row or column. Most particularly, matrix enlarging can results in a reduction of the fill-in in theR matrix which occurs during orthogonal factorization as a result of dense rows. Numerical experiments support these conjectures.     

Alternating sign matrices and descending plane partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1, ?1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math.53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math.66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.