On the complete pivoting conjecture for Hadamard matrices: further progress and a good pivots property

Further progress is achieved for the growth conjecture for Hadamard matrices. It is proved that the leading principal minors of a CP Hadamard matrix form an increasing sequence. Bounds for the sixth and seventh pivot of any CP Hadamard matrix are given. A new proof demonstrating that the growth of a Hadamard matrix of order 12 is 12, is presented. Moreover, a new notion of good pivots is introduced and its importance for the study of the growth problem for CP Hadamard matrices is examined. We establish that CP Hadamard matrices with good pivots satisfy Cryer’s growth conjecture with equality, namely their growth factor is equal to their order. A construction of an infinite class of Hadamard matrices is proposed.     

On reconstruction of a matrix by its minors

In this paper, we reconstruct matrices from their minors, and give explicit formulas for the reconstruction of matrices of orders 2 × 3, 2 × 4, 2 × n, 3 × 6 and m × n. We also formulate the Plücker relations, which are the conditions of the existence of a matrix related to its given minors.     

A counterexample to a hadamard matrix pivot conjecture

In the study of the growth factor of completely pivoted Hadamard matrices, it becomes natural to study the possible pivots. Very little is known or provable about these pivots other than a few cases at the beginning and end. For example it is known that the first four pivots must be 1,2,2 and 4 and the last three pivots in backwards order must be n/2, and n/2. Based on computational experiments, it was conjectured by Day and Peterson, that the n—3rd pivot must always be n/4. This conjecture would have suggested a kind of symmetry with the first four pivots. In this note we demonstrate a matrix whose n-3rd pivot is n/2 showing that the conjecture is false.     

Nonnegativity of principal minors of generalized inverses of M-matrices

The result of principal interest established in this paper is that if A is an n × n singular irreducible M-matrix, then a large class of generalized inverses of A possesses the property that each of its elements has all its principal minors nonnegative. The class contains both the group and the Moore-Penrose generalized inverses of A. In an application of our results it is shown that the fundamental matrix of a continuous (in time) ergodic Markov chain on a finite state space has all its principal minors nonnegative.     

D-optimal designs embedded in Hadamard matrices and their effect on the pivot patterns

In this paper we develop a new approach for detecting if specific D-optimal designs exist embedded in Sylvester-Hadamard matrices. Specifically, we investigate the existence of the D-optimal designs of orders 5, 6, 7 and 8. The problem is motivated to explaining why specific values appear as pivot elements when Gaussian elimination with complete pivoting is applied to Hadamard matrices. Using this method and a complete search algorithm we explain, for the first time, the appearance of concrete pivot values for equivalence classes of Hadamard matrices of orders n = 12, 16 and 20.     

Factorizing symmetric indefinite matrices

The LDL^T factorization of a symmetric indefinite matrix, although efficient computationally, may not exist and can be unstable in the presence of round off error. The use of block diagonal 2×2 pivots is attractive, but there are some difficulties in determining an efficient and stable pivot strategy. Previous suggestions have required O(n>³) operations (either multiplications or comparisons) just to implement the pivot strategy. A new strategy is described which in practice only requires O(n²) operations. Indeed, the effort required by this pivot strategy is less than that required when using partial pivoting with an unsymmetric LU factorization, which is the usual way of factorizing indefinite matrices.     

New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to simultaneously enter into the basis. We call this operation a multiple pivot. We show that a shortest path simplex algorithm with this pivot rule performs O(n) multiple pivots and runs in O(nm) time. Our pivot rule is based on the new concept of a pseudo permanently labeled node, and it can be adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept lets us introduce new rules to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.     

Smith meets Smith: Smith normal form of Smith matrix

In 1861, Henry John Stephen Smith [H.J.S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans. Royal Soc. London. 151 (1861), pp. 293–326] published famous results concerning solving systems of linear equations. The research on Smith normal form and its applications started and continues. In 1876, Smith [H.J.S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875/76), pp. 208–212] calculated the determinant of the n?×?n matrix ((i,?j)), having the greatest common divisor (GCD) of i and j as its ij entry. Since that, many results concerning the determinants and related topics of GCD matrices, LCM matrices, meet matrices and join matrices have been published in the literature. In this article these two important research branches developed by Smith, in 1861 and in 1876, meet for the first time. The main purpose of this article is to determine the Smith normal form of the Smith matrix ((i,?j)). We do this: we determine the Smith normal form of GCD matrices defined on factor closed sets.     

A pivoting strategy for symmetric tridiagonal matrices

The LBL^T factorization of Bunch for solving linear systems involving a symmetric indefinite tridiagonal matrix T is a stable, efficient method. It computes a unit lower triangular matrix L and a block 1 × 1 and 2 × 2 matrix B such that T=LBL^T. Choosing the pivot size requires knowing a priori the largest element σ of T in magnitude. In some applications, it is required to factor T as it is formed without necessarily knowing σ. In this paper, we present a modification of the Bunch algorithm that can satisfy this requirement. We demonstrate that this modification exhibits the same bound on the growth factor as the Bunch algorithm and is likewise normwise backward stable. Copyright © 2005 John Wiley & Sons, Ltd.     

A new pivot selection rule for the network simplex algorithm

We present a new network simplex pivot selection rule, which we call theminimum ratio pivot rule, and analyze the worst-case complexity of the resulting network simplex algorithm. We consider networks withn nodes,m arcs, integral arc capacities and integral supplies/demands of nodes. We define a {0, 1}-valued penalty for each arc of the network. The minimum ratio pivot rule is to select that eligible arc as the entering arc whose addition to the basis creates a cycle with the minimum cost-to-penalty ratio. We show that the so-defined primal network simplex algorithm solves minimum cost flow problem within O(nΔ) pivots and in O(Δ(m + n logn)) time, whereΔ is any upper bound on the sum of all arc flows in every feasible flow. For assignment and shortest path problems, our algorithm runs in O(n ²) pivots and O(nm +n ² logn) time.     

Improving Hadamard's inequality *

We study several bounds for the determinant of an n × n positive definite Hermitian matrix A. These bounds are the best possible given certain data about A. We find the best bounds in the cases that we are given: (i) the diagonal elements of A: (ii) the traces trA,tr A ² and n and (iii)n, tr A tr A ² and the diagonal elements of A. In case (i) we get the well known Hadamard inequality. The other bounds are Hadamard type bounds. The bounds are found using optimization techniques.     

Rang des matrices Hermitiennes semi-définies positives indéfiniment divisibles

In this paper we study n × n Hermitian semidefinite positive matrices which are infinitely divisible in a sense that we define in Sec. 1. We establish (Theorem 2.2) a stability property for the rank of the “Hadamard power matrices” of such a matrix.     

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Let a, b and c be fixed complex numbers. Let M _n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M _n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M _n (a, b, c). We also show that all complex polynomials in M _n (a, b, c) are Toeplitz matrices. In particular, the inverse of M _n (a, b, c) is a Toeplitz matrix when it exists.     

Ideals of generic minors

In this paper we study the well-known example of the ideal generated by all the r × r minors in the generic n × m matrix (X_ij) from the point of view of Gröebner bases.     

Distance matrices for points on a line,on a circle,and at the vertices of ann-dimensional cube

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ ^m , the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995.     

On inverse multiplicative eigenvalue problems for matrices

Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ₁,…,λ_n} in C. The number of different matrices D is at most n!.     

On convertible nonnegative matrices * †

An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A= det(H°A) where H ° Adenotes the Hadamard product of Hand A. A convertible (0,l)-matrix is called extremal if replacing any zero entry with a 1 breaks the convertibility. In this paper some properties of

nonnegative convertible matrices are investigated and some classes of extremal convertible (0,1)-matrices are obtained.     

A matrix of permanents

If A is a matrix of order n×(n?2), n?3, denote by ā the n×n matrix whose (i,j)th entry is zero if i=j, and if i≠j, is the permanent of the submatrix of A obtained by deleting its ith and jth rows. It is shown that if A is a nonnegative n×(n?2) matrix, then ā is nonsingular if and only if A has no zero submatrix of n?1 lines. This is used to give precise consequences of the occurrence of equality in Alexandroff's inequality.     

The inverse problem of bisymmetric matrices with a submatrix constraint

An n×n real matrix A is called a bisymmetric matrix if A=A^T and A=S_nAS_n, where S_n is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A₀, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A^*, find an n×n matrix  in S_E such that where ∥·∥ is Frobenius norm, and S_E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.