Extensions of band matrices with band inverses

Let R_ij be a given set of μ_i× μ_j matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[F_ij], i,j=1,…, n, such that F_ij=R_ij for |i?j|?m, F admits either a left or right block triangular factorization, and (F^?1)_ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given R_ij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied.     

On constant discrete programming problems

Let H be a subset of the set S_n of all permutations

$\text{1}\text{2}\text{???}\text{n}\text{s(1)}\text{s(2)}\text{???}\text{s(n)}$

C=6c_ij6 a real n?n matrix L_c(s)=c_1s(1)+c_2s(2)+???+c_ns(n) for s ? H. A pair (H, C) is the existencee of reals a₁,b₁,a₂,b₂,…a_n,b_n, for which c_ij=a₁+b_j if (i,j)?D(H), where D(H)={(i,j):(?h?H)(j=h(i))}.For a pair (H,C) the specifity of it is proved in the case, when H is either a special cyclic class of permutations or a special union of cyclic classes. Specific pairs with minimal sets H are in some sense described.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

On orthogonal matrices with constant diagonal

In connection with the problem of finding the best projections of k-dimensional spaces embedded in n-dimensional spaces Hermann König asked: Given m∈$\text{R}$ and n∈$\text{N}$, are there n×n matrices C=(c_ij), i, j=1,…,n, such that c_ii=m for all i, |c_ij|=1 for i≠j, and C²=(m²+n?1)I_n? König was especially interested in symmetric C, and we find some families of matrices satisfying this condition. We also find some families of matrices satisfying the less restrictive condition CC^T=(m²+n?1)I_n.     

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? $\text{R}$^m, v?$\text{R}$ⁿ, find an (m,n) matrix X ? 0 with prescribed row sums Σⁿ_j=1X_ij = u_i (i = 1,…,m) and prescribed column sums Σ^m_i=1X_ij = v_j (j = 1,…,n) which fits the relations X_ij = A_ij + λ_iW_ij + W_ij + W_ijμ_j for all i,j and some λ?$\text{R}$^m, μ?$\text{R}$ⁿ. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].     

Some explicit formulae in simultaneous padé approximation

Using old results on the explicit calculation of determinants, formulae are given for the coefficients of P₀(z) and P₀(z)f_i(z) ? P_i(z), where P_i(z) are polynomials of degree σ ? ρ_i (i=0,1,…,n), P₀(z)f_i(z) ? P_i(z) are power series in which the terms with z^k, 0?k?σ, vanish (i=1,2,…,n), (ρ₀,ρ₁,…,ρ_n) is an (n+1)-tuple of nonnegative integers, σ=ρ₀+ρ₁+?+ρ_n, and {f_i}ⁿ_i=1 is the set of hypergeometric functions {₁F₁(1;c_i;z)}ⁿ_i=1$\text{(c}{}_{\mathrm{i}}\text{?}\text{Z}\text{z.drule;}\text{N}\text{, c}{}_{\mathrm{i}}\text{? c}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ or {₂F₀(a_i,1;z)}ⁿ_i=1$\text{(a}{}_{\mathrm{i}}\text{?}\text{Z}\text{?}\text{N}\text{, a}{}_{\mathrm{i}}\text{? a}{}_{\mathrm{j}}\text{?}\text{Z}\text{)}$ under the condition ρ₀?ρ_i ? 1 (i=1,2,…,n).     

Canonical row-column-exchangeable arrays

Consider a standard row-column-exchangeable array X = (X_ij : i,j ≥ 1), i.e., X_ij = f(a, ξ_i, η_j, λ_ij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ′₁, ξ′₂,…, η′₁, η′₂,…, are conditionally independent given ∩_{n ≥ 1}σ(X′_ij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous.     

Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)²=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA _n ^c )A _n is introduced: It hasn?1 generatorss ₁,…,S _n?2,e, theA _n-relations (above), and relationse ²=e, (es ₁)⁴, (es _j)²=(es _j)⁴,es _i=s _i s ₁ ^-1 es ₁, wherej>1 andi≥1.     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Class G,and a unification problem

In 1958, Karl Goldberg proved the following: Theorem G. Suppose A=(a_ij) is an n×n matrix over the complex field, with the following properties: (1) a_ija_ji?0 for i,j=1,2,…,n, and (2) a_i1i2a_i2i3?a_isii=a_i2i1a_i3i2?a_i1is for all s=1,2,…,n and i_t=1,2,…,n. Then A has only real characteristic values. Definition. Let G_n denote the class of n×n matrices over $\text{C}$, the complex field, which satisfy the Goldberg conditions (1) and (2). We investigate some properties of class G related to the following topics: Schur complements, weak sign symmetry, and inequalities due to Oppenheim for positive definite matrices, and an analogue due to Markham for tridiagonal, oscillatory matrices.     

A Fisher type inequality

In this paper we study subsets of a finite set that intersect each other in at most one element. Each subset intersects most of the other subsets in exactly one element. The following theorem is one of our main conclusions. Let S₁,… S_m be m subsets of an n-set S with |S₁| ? 2 (l = 1, …,m) and |S_i ∩ S_j| ? 1 (i ≠ j; i, j = 1, …, m). Suppose further that for some fixed positive integer c each S_i has non-empty intersection with at least m ? c of the remaining subsets. Then there is a least positive integer M(c) depending only on c such that either m ? n or m ? M(c).     

Sorting by means of swappings

If π is a permutation of {1,…,n}, then the effect of the swap S_ij on π is that the ith and fth entry are sorted. If [i?j] = l, we call S_ij a miniswap (it sorts neighbours). The paper studies a partial order based on the swaps, and establishes some properties of miniswaps.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

The Jordan canonical form for a class of zero-one matrices

Let f:N→N be a function. Let A_n=(a_ij) be the n×n matrix defined by a_ij=1 if i=f(j) for some i and j and a_ij=0 otherwise. We describe the Jordan canonical form of the matrix A_n in terms of the directed graph for which A_n is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture.     

The inertia of a Hermitian matrix having prescribed diagonal blocks

For i=1,…,m let H_i be an n_i×n_i Hermitian matrix with inertia In(H_i)= (π_i, ν_i, δ_i). We characterize in terms of the π_i, ν_i, δ_i the range of In(H) where H varies over all Hermitian matrices which have a block decomposition H= (X_ij)_{i, j=1,…, m} in which X_ij is n_i×n_j and X_ii=H_i.     

Spectrum of Ornstein-Uhlenbeck Operators in L Spaces with Respect to Invariant Measures

Let A=∑_i,j=1^Nq_ijD_ij+∑_i,j=1^Nb_ijx_jD_i be a possibly degenerate Ornstein-Uhlenbeck operator in R^N and assume that the associated Markov semigroup has an invariant measure μ. We compute the spectrum of A in L_μ^p for 1?p<∞.     

On discount residue classes

It is shown that, whenever m₁, m₂,…, m_n are natural numbers such that the pairwise greatest common divisors, d_ij=(m_i, m_j), i≠j are distinct and different from 1, then there exist integers a₁, a₂,…,a_n such that the solution sets of the congruences x≡_i (modm_i), i= 1,2,…,n are disjoint.     

Sequences of orthogonal vectors with a fixed norm

Let A = (a_ij) be an n × m matrix with a_ij ∈ K, a field of characteristic not 2, where Σ_i=1ⁿa_ij² = e, 1 ≤ j ≤ m, and Σ_i=1ⁿa_ija_ij′ = 0 for j ≠ j′. The question then is when is it possible to extend A, by adding columns, to obtain a matrix with orthogonal columns of the same norm. The question is answered for n ? 7 ≤ m ≤ n as well as for more general cases. Complete solutions are given for global and local fields, the answer depending on what congruence class modulo 4 n belongs to and how few squares are needed to sum to e.     

On bounds for eigenvalues of real symmetric matrices

Let A=(a_ij) be a real symmetric matrix of order n. We characterize all nonnegative vectors x=(x₁,...,x_n) and y=(y₁,...,y_n) such that any real symmetric matrix B=(b_ij), with b_ij=a_ij, i≠jhas its eigenvalues in the union of the intervals [b_ij?y_i, b_ij+ x_i]. Moreover, given such a set of intervals, we derive better bounds for the eigenvalues of B using the 2n quantities {b_ii?y, b_ii+x_i}, i=1,..., n.