Geometric conditions for the existence of positive eigenvalues of matrices

Finite-dimensional theorems of Perron-Frobenius type are proved. For A∈$\text{C}$ⁿⁿ and a nonnegative integer k, we let w_k (A) be the cone generated by A^k, A^k+1,…in $\text{C}$ⁿⁿ. We show that A satisfies the Perron-Schaefer condition if and only if the closure $\text{W}$_k(A) of w_k(A) is a pointed cone. This theorem is closely related to several known results. If k?v₀(A), the index of the eigenvalue 0 in spec A, we prove that A has a positive eigenvalue if and only if w_k(A) is a pointed nonzero cone or, equivalently $\text{W}$_k(A) is not a real subspace of $\text{C}$ⁿⁿ. Our proofs are elementary and based on a method of Birkhoff's. We discuss the relation of this method to Pringsheim's theorem.     

Power inequalities and spectral dominance of generalized matrix norms

A generalized matrix norm G dominates the spectral radius for all A?M_n(C) (i) if for some positive integer k the rule G(A^k) ? G(A)^k holds for all A?M_n(C) and (ii) if and only if for each A?M_n(C) there exists a constant γ_A such that G(A^k) ? γ_AG(A)^kfor all positive integers k. Other results and examples are also given concerning spectrally dominant generalized matrix norms.     

Obstructions to embeddings of bundles of matrix algebras in a trivial bundle

We evaluate the cohomology obstructions to the existence of fiber-preserving unital embedding of a locally trivial bundle A _k → X whose fiber is a complex matrix algebra M _k (?) in a trivial bundle with fiber M _kl (?) under the assumption that (k, l) = 1. It is proved that the first obstruction coincides with the obstruction to the reduction of the structure group PGL _k (?) of the bundle A _k to SL _k (?), which coincides with the first Chern class c ₁(ξ _k ) reduced modulo k under the assumption that A _k ≌ End(ξ _k ) for some vector ? _k -bundle ξ _k → X. If the first obstruction vanishes, then A _k ≌ End( $\tilde \xi _k $ ) for some vector ? ^k bundle ξ _k → X with structure group SL _k (?), and the second obstruction is c ₂( $\tilde \xi _k $ )modk ∈ H ⁴(X, ?/k?). Further, the higher obstructions are defined using a Postnikov tower, and each of the obstructions is defined on the kernel of the previous obstruction.     

An oblique projection iterative method to compute Drazin inverse and group inverse

In this paper, an oblique projection iterative method is presented to compute matrix equation AXA=A of a square matrix A with ind(A)=1. By this iterative method, when taken the initial matrix X₀=A, the group inverse A_g can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation A^kXA^k=A^k, a group inverse (A^k)_g of matrix A^k is got, then we use the formulae A_d=A^k-1(A^k)_g, the Drazin inverse A_d can be obtained.     

The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc

We consider the complex differential equations of the form

Ak(z)f(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=F(z),     

A noncommutative analogue of |D(Xk|=|kXk−1|

The paper concerns alternating powers of a Hilbert space. Let ∧^k be defined by ∧^k(A)(x₁∧?∧x_k)=Ax₁∧?∧Ax_k. It is proved that the norm of the linear map D∧^k(A) depends only upon |A| and is assumed at the identity.     

Self-dual and quasi self-dual algebras

Self-dual algebras are ones with an A bimodule isomorphism A → A ^∨op, where A ^∨ = Hom _k (A, k) and A ^∨op is the same underlying k-module as A ^∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*(A,A) ≌ H*(A,A ^{∨ op}). For all such algebras H*(A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual.     

Defect indices of powers of a contraction

Let A be a contraction on a Hilbert space H. The defect index d_A of A is, by definition, the dimension of the closure of the range of I-A^∗A. We prove that (1) d_Aⁿ?nd_A for all n?0, (2) if, in addition, Aⁿ converges to 0 in the strong operator topology and d_A=1, then d_Aⁿ=n for all finite n,0?n?dimH, and (3) d_A=d_A^∗ implies d_Aⁿ=d_A^n∗ for all n?0. The norm-one index k_A of A is defined as sup{n?0:‖Aⁿ‖=1}. When dimH=m<∞, a lower bound for k_A was obtained before: k_A?(m/d_A)-1. We show that the equality holds if and only if either A is unitary or the eigenvalues of A are all in the open unit disc, d_A divides m and d_Aⁿ=nd_A for all n, 1?n?m/d_A. We also consider the defect index of f(A) for a finite Blaschke product f and show that d_f(A)=d_Aⁿ, where n is the number of zeros of f.     

Deletion-restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A₀. Let A^′=A?{A₀} be the deletion of A₀ from A and A^″={A∩A₀|A∩A₀≠∅} be the restriction of A to A₀. Let M=V?_?A∈AA be the complement of A in V. If A is an arrangement of complex affine hyperplanes, then there is a split short exact sequence, 0→Hk(M^′)→Hk(M)→Hk+1−codimR(A₀)(M^″)→0. In this paper, we determine conditions for when the triple (A,A^′,A^″) of arrangements of affine subspaces yields the above split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion-restriction subspace arrangements.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

A generalization of Lavoie's inequality concerning the sum of idempotent matrices

Let a complex pxn matrix A be partitioned as A′=(A′₁,A′₂,…,A′_k). Denote by ?(A), A′, and A^? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A⁽¹⁴⁾ be an inner inverse of A such that A⁽¹⁴⁾A is a Hermitian matrix. Let B=(A⁽¹⁴⁾₁,A⁽¹⁴⁾₂,…,A_k⁽¹⁴⁾) and $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$.Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A⁽¹⁴⁾ or $\text{A}{}_{\mathrm{i>}}\text{A}{}_{\mathrm{<mtext>j</mtext>}}{}^1\text{=0}$ for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$ is no longer true.     

Blocker sets, orthogonal arrays and their application to combination locks

Let A denote a set of order m and let X be a subset of A^k+1. Then X will be called a blocker (of A^k+1) if for any element say (a₁,a₂,…,a_k,a_k+1) of A^k+1, there is some element (x₁,x₂,…,x_k,x_k+1) of X such that x_i equals a_i for at least two i. The smallest size of a blocker set X will be denoted by α(m,k)and the corresponding blocker set will be called a minimal blocker. Honsberger (who credits Schellenberg for the result) essentially proved that α(2n,2) equals 2n² for all n. Using orthogonal arrays, we obtain precise numbers α(m,k) (and lower bounds in other cases) for a large number of values of both k and m. The case k=2 that is three coordinate places (and small m) corresponds to the usual combination lock. Supposing that we have a defective combination lock with k+1 coordinate places that would open if any two coordinates are correct, the numbers α(m,k) obtained here give the smallest number of attempts that will have to be made to ensure that the lock can be opened. It is quite obvious that a trivial upper bound for α(m,k) is m² since allowing the first two coordinates to take all the possible values in A will certainly obtain a blocker set. The results in this paper essentially prove that α(m,k) is no more than about m²/k in many cases and that the upper bound cannot be improved. The paper also obtains precise values of α(m,k) whenever suitable orthogonal arrays of strength two (that is, mutually orthogonal Latin squares) exist.     

Adequate equivalence relations and Pontryagin products

Let A be an abelian variety over a field k. We consider CH₀(A) as a ring under Pontryagin product and relate powers of the ideal I⊆CH₀(A) of degree zero elements to powers of the algebraic equivalence relation. We also consider a filtration F⁰⊇F¹⊇… on the Chow groups of varieties of the form T×_kA (defined using Pontryagin products on A×_kA considered as an A-scheme via projection on the first factor) and prove that F^r coincides with the r-fold product (F¹)^*r as adequate equivalence relations on the category of all such varieties.     

Accurate approximation of time-varying matrices of linear differential systems

We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation A_n,k = ∑ⁿ_{j ? 1}c_j{Y(t_k + τ_j) Y^?1(t_k) ? I} to A(t_k) for grid points t_k = a + kh, k = 0,…, N using specified τ_j = θ_jh, 0 < θ_j < 1, j = 1, …, n, and show that for each t_k, the L₁ norm of the error matrix is $\text{O}$(hⁿ). We demonstrate an efficient scheme for the evaluation of A_n,k and treat sample problems.     

Disjoint matchings of graphs

In this paper, we prove a generalization of the familiar marriage theorem. One way of stating the marriage theorem is: Let G be a bipartite graph, with parts S₁ and S₂. If A ? S₁ and F(A) ? S₂ is the set of neighbors of points in A, then a matching of G exists if and only if Σ_x∈S2 min(1, | F^?1(x) ∩ A |) ≥ | A | for each A ? S₁. Our theorem is that k disjoint matchings of G exist if and only Σ_x∈S2 min (k, | F^?1(x) ∩ A |) ≥ k | A | for each A ? S₁.     

On D. Hägele’s approach to the Bessis-Moussa-Villani conjecture

The reformulation of the Bessis-Moussa-Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient α_m,k(A,B) of t^k in the polynomial Tr(A+tB)^m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd.