Geršgorin variations III: On a theme of Brualdi and Varga

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi’s conditions are sharp. Improving Varga’s results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.     

The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids Γ, the E-theoretic descent functor at the C^∗-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. Section 2 shows that Γ-actions on a C₀(X)-algebra B, where X is the unit space of Γ, can be usefully formulated in terms of an action on the associated bundle B^?. Section 3 shows that the functor B→C^∗(Γ,B) is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ, the main technical difficulty involved being that of finding a Γ-algebra that plays the role of Cbcont(T,B) in the group case.     

Geršgorin type theorems for quaternionic matrices

This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.     

On entropy-preserving stochastic averages

When an n×n doubly stochastic matrix A acts on Rⁿ on the left as a linear transformation and P is an n-long probability vector, we refer to the new probability vector AP as the stochastic average of the pair (A,P). Let Γ_n denote the set of pairs (A,P) whose stochastic average preserves the entropy of P:H(AP)=H(P). Γ_n is a subset of B_n×Σ_n where B_n is the Birkhoff polytope and Σ_n is the probability simplex.We characterize Γ_n and determine its geometry, topology,and combinatorial structure. For example, we find that (A,P)∈Γ_n if and only if A^tAP=P. We show that for any n, Γ_n is a connected set, and is in fact piecewise-linearly contractible in B_n×Σ_n. We write Γ_n as a finite union of product subspaces, in two distinct ways. We derive the geometry of the fibers (A,·) and (·,P) of Γ_n. Γ₃ is worked out in detail. Our analysis exploits the convexity of xlogx and the structure of an efficiently computable bipartite graph that we associate to each ds-matrix. This graph also lets us represent such a matrix in a permutation-equivalent, block diagonal form where each block is doubly stochastic and fully indecomposable.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

Vanishing viscosity in the plane for vorticity in borderline spaces of Besov type

The existence and uniqueness of solutions to the Euler equations for initial vorticity in BΓ∩Lp₀∩Lp₁ was proved by Misha Vishik, where BΓ is a borderline Besov space parameterized by the function Γ and 1<p₀<2<p₁. Vishik established short time existence and uniqueness when Γ(n)=O(logn) and global existence and uniqueness when . For initial vorticity in BΓ∩L², we establish the vanishing viscosity limit in L²(R²) of solutions of the Navier-Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ(n)=O(logn) and uniform over any finite time when Γ(n)=O(logκn), 0?κ<1, and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include BΓ∩L² when Γ(n)=O(logκn) for 0<κ<1.     

A note on k-potence preserving maps

Let M_n be the algebra of all n×n complex matrices and Γ_n the set of all k-potent matrices in M_n. Suppose ?:M_n→M_n is a map satisfying A-λB∈Γ_n implies ?(A)-λ?(B)∈Γ_n, where A, B∈M_n, λ∈C. Then either ? is of the form ?(A)=cTAT^-1, A∈M_n, or ? is of the form ?(A)=cTA^tT^-1, A∈M_n, where T∈M_n is an invertible matrix, c∈C satisfies c^k=c.     

Complementation and decompositions in some weakly Lindelöf Banach spaces

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c₀(Γ) or (2) for every c₀(Γ)⊆X has a complemented c₀(E) for an uncountable E⊆Γ or (3) has a decomposition X=A⊕B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P-ideal dichotomy, for every c₀(Γ)⊆C(K) there is a complemented c₀(E) for an uncountable E⊆Γ, which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C(K) for K of height ω+1 where every operator is of the form cI+S for c∈R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K).     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(M_n(D)) and determine the diameters of some induced subgraphs of Γ(M_n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M_n(D). For every field F, it is shown that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 6. We conjecture that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

Metric characterization of apartments in dual polar spaces

Let Π be a polar space of rank n and let Gk(Π), k∈{0,…,n−1} be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γk(Π). We consider the polar Grassmannian Gn−1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γn−1(Π) is an apartment of Gn−1(Π). This follows from a more general result concerning isometric embeddings of Hm, m?n in Γn−1(Π). As an application, we classify all isometric embeddings of Γn−1(Π) in Γn^′−1(Π^′), where Π^′ is a polar space of rank n^′?n.     

H-matrix theory vs. eigenvalue localization

The eigenvalue localization problem is very closely related to the -matrix theory. The most elegant example of this relation is the equivalence between the Geršgorin theorem and the theorem about nonsingularity of SDD (strictly diagonally dominant) matrices, which is a starting point for further beautiful results in the book of Varga [19]. Furthermore, the corresponding Geršgorin-type theorem is equivalent to the statement that each matrix from a particular subclass of -matrices is nonsingular. Finally, the statement that all eigenvalues of a given matrix belong to minimal Geršgorin set (defined in [19]) is equivalent to the statement that every -matrix is nonsingular. Since minimal Geršgorin set remained unattainable, a lot of different Geršgorin-type areas for eigenvalues has been developed recently. Along with them, a lot of new subclasses of -matrices were obtained. A survey of recent results in both areas, as well as their relationships, will be presented in this paper.This work is partly supported by Republic of Serbia, Ministry of Science and Environmental Protection, grant 144025.     

Representations for the weighted Drazin inverse of a sum of matrices

Three representations for the W-weighted Drazin inverse of a matrix A?CWB have been developed under some conditions where A,B,C∈? ^m×n , and W∈? ^n×m . The results of this paper not only extend the earlier works about the Drazin inverse and group inverse, but also weaken the assumed condition of a result of the Drazin inverse to the case where Γ _d ZZ _g =ZZ _g Γ _d is substituted with C(Γ _d ZZ _g ?ZZ _g Γ _d )B=0. Numerical examples are given to illustrate some new results.     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

On weakly unitarily invariant norm and the λ-Aluthge transformation for invertible operator

Let T∈B(H) be an invertible operator with polar decomposition T = UP and B∈B(H) commute with T. In this paper we prove that ∣∣∣P^λBUP^1−λ∣∣∣ ? ∣∣∣BT∣∣∣, where ∣∣∣ · ∣∣∣ is a weakly unitarily invariant norm on B(H) and 0 ? λ ? 1. As the consequence of this result, we have ∣∣∣f(P^λUP^1−λ)∣∣∣ ? ∣∣∣f(T)∣∣∣ for any polynomial f.     

On upper domination Ramsey numbers for graphs

The classical Ramsey number r(m,n) can be defined as the smallest integer p such that in every two-coloring (R,B) of the edges of K_p, β(B)⩾m or β(R)⩾n, where β(G) denotes the independence number of a graph G. We define the upper domination Ramsey number u(m,n) as the smallest integer p such that in every two-coloring (R,B) of the edges of K_p, Γ(B)⩾m or Γ(R)⩾n, where Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. The mixed domination Ramsey number v(m,n) is defined to be the smallest integer p such that in every two-coloring (R,B) of the edges of K_p, Γ(B)⩾m or β(R)⩾n. Since β(G)⩽Γ(G) for every graph G, u(m,n)⩽v(m,n)⩽r(m,n). We develop techniques to obtain upper bounds for upper domination Ramsey numbers of the form u(3,n) and mixed domination Ramsey numbers of the form v(3,n). We show that u(3,3)=v(3,3)=6, u(3,4)=8, v(3,4)=9, u(3,5)=v(3,5)=12 and u(3,6)=v(3,6)=15.