Combining triple diagonal forms

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R-matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. Two different refinements have since been made to Jacobi’s triple diagonal form. This paper works toward combining these refinements.     

Sharpening the triple diagonal form

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R-matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. This paper focuses on controlling the off-diagonal elements.     

Cyclic and diagonal products on a matrix

We unify the theory of cyclic and diagonal products of elements of matrices. We obtain some new results on diagonal similarity, diagonal equivalence, complete reducibility and total support.     

Cones, graphs and optimal scalings of matrices

Characterizations are given of the optimal scalings of a complex square matrix within its diagonal similarity class and its restricted diagonal equivalence class with respect to the maximum element norm. The characterizations are in terms of a finite number of products, principally circuit and diagonal products. The proofs proceed by reducing the optimal scaling problems from the multiplicative matrix level in succession to an additive matrix level, a graph theoretic level, and a geometric level involving duality theorems for cones. At the geometric level, the diagonal similarity and the restricted diagonal equivalence problems are unified.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

The α-scalar diagonal stability of block matrices

We generalize two results: Kraaijevanger’s 1991 characterization of diagonal stability via Hadamard products and the block matrix version of the closure of the positive definite matrices under Hadamard multiplication. We restate our generalizations in terms of P^α-matrices and α-scalar diagonally stable matrices.     

Orbifolds and groupoids

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases on (Pre-Orb) and Morita equivalence in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.     

Hasse-Arf Property and Abelian Extensions

The paper contains discussions of relations between the property of a totally ramified p-extension of a local field to be abelian and the property of its Galois group to possess integer jumps with respect to the upper numbering (Hasse-Arf property). It is shown that such an extension L/F is abelian if and only if for any totally ramified abelian extension E/F the extension LE/F satisfies the Hasse-Arf property. An additional property to the Hasse-Arf property in terms of principal units which makes the extension abelian is established as well.     

Graphically abelian groups

We define a group G to be graphically abelian if the function g?g⁻¹ induces an automorphism of every Cayley graph of G. We give equivalent characterizations of graphically abelian groups, note features of the adjacency matrices for Cayley graphs of graphically abelian groups, and show that a non-abelian group G is graphically abelian if and only if G=E×Q, where E is an elementary abelian 2-group and Q is a quaternion group.     

Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.     

Matrices with isomorphic diagonal hypergraphs

Let A be an n×n matrix of 0's and 1's with total support. The diagonal hypergraph of A is the hypergraph whose vertices are the positive positions of A and whose edges are the positive diagonals of A. We investigate which n×n matrices with total support have diagonal hypergraphs isomorphic to that of A, and the structure of such isomorphisms.     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

On determinantal diagonal dominance conditions

The paper suggests sufficient nonsingularity conditions for matrices in terms of certain determinantal relations of diagonal dominance type, which improve and generalize some known results. These conditions are used to describe new eigenvalue inclusion sets and to derive new two-sided bounds on the determinants of matrices satisfying them. Bibliography: 8 titles.     

Schur-type stability properties and complex diagonal scaling

We introduce new classes of n-by-n matrices with complex entries which can be scaled by a diagonal matrix with complex entries to be normal or Hermitian and study the Schur-type stability properties of these matrices.     

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.     

Resonance in equations with a diagonal field

Let A be a real square matrix, and let J?$\text{R}$ be an interval not containing an eigenvalue of A. Is A–D nonsingular for all diagonal matrices D with entries d_i ∈J? This holds if A is symmetric, but is not true in general. We prove a necessary condition and indicate implications for an equation with a diagonal field.     

A note on generalizations of strict diagonal dominance for real matrices

We investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M- matrices.     

Identities among certain triangular matrices

The object of this paper is to develop the ideas introduced in the author's paper [1] on matrices which generate families of polynomials and associated infinite series. A family of infinite one-subdiagonal non-commuting matrices Q_m is defined, and a number of identities among its members are given. The matrix Q₁ is applied to solve a problem concerning the derivative of a family of polynomials, and it is shown that the solution is remarkably similar to a conventional solution employing a scalar generating function. Two sets of infinite triangular matrices are then defined. The elements of one set are related to the terms of Laguerre, Hermite, Bernoulli, Euler, and Bessel polynomials, while the elements of the other set consist of Stirling numbers of both kinds, the two-parameter Eulerian numbers, and numbers introduced in a note on inverse scalar relations by Touchard. It is then shown that these matrices are related by a number of identities, several of which are in the form of similarity transformations. Some well-known and less well-known pairs of inverse scalar relations arising in combinatorial analysis are shown to be derivable from simple and obviously inverse pairs of matrix relations. This work is an explicit matrix version of the umbral calculus as presented by Rota et al. [24-26].