Abstract: Publication date: Available online 12 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Froilán M. Dopico, Vanni Noferini We develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials, i.e., either regular or singular, and study how these vector polynomials are related to the spectral properties of matrix polynomials. We pay particular attention to the so-called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues (finite or infinite) of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its Smith form. In addition, we describe clearly, for the first time in the literature, the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade. Once the main theoretical properties of root polynomials have been established, the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed. More precisely, we study the change of root polynomials under rational transformations, or reparametrizations, of matrix polynomials, the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations, and the relationship between the root polynomials of two dual pencils. We emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of Jordan chains, as a consequence of the well known relationship between root polynomials and Jordan chains. Therefore, a number of open problems are also solved for Jordan chains of regular matrix polynomials. We also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials.
Abstract: Publication date: Available online 12 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Olga Limantseva, George Halikias, Nicos Karcanias The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations.
Abstract: Publication date: Available online 5 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Francesco Galuppi We study signature tensors of paths from an algebraic geometric viewpoint. The signatures of a given class of paths parametrize a variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths and new challenging questions about their behavior. This paper focuses on signatures of rough paths. Their signature variety shows surprising analogies with the Veronese variety, and our aim is to prove that this so-called Rough Veronese is toric. The same holds for the universal variety. Answering a question of Améndola, Friz and Sturmfels, we show that the ideal of the universal variety does not need to be generated by quadrics.
Abstract: Publication date: Available online 5 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): M.A. Goberna, J.E. Martínez-Legaz, M.I. Todorov Given an arbitrary set T in the Euclidean space Rn, whose elements are called sites, and a particular site s, the farthest Voronoi cell of s, denoted by FT(s), consists of all points which are farther from s than from any other site. In this paper we study farthest Voronoi cells and diagrams corresponding to arbitrary (possibly infinite) sets. More in particular, we characterize, for a given arbitrary set T, those s∈T such that FT(s) is nonempty and study the geometrical properties of FT(s) in that case. We also characterize those sets T whose farthest Voronoi diagrams are tesselations of the Euclidean space, and those sets that can be written as FT(s) for some T⊂Rn and some s∈T.
Abstract: Publication date: Available online 5 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Ben W. Grossmann, Hugo J. Woerdeman We characterize the linear maps that preserve maximally entangled states in L(X⊗Y) in the case where dim(X) divides dim(Y).
Abstract: Publication date: Available online 5 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Constantin Costara Let F be a field having at least n2+1 distinct elements, and denote by Mn the space of all n×n matrices over F. We prove that if φ and ψ are maps from Mn into itself, one of them being surjective, such that det(φ(x)+ψ(y))=det(x+y) for all x,y∈Mn, there exist then x0∈Mn and u,v∈Mn satisfying det(uv)=1 such that either φ(x)=u(x+x0)v and ψ(x)=u(x−x0)v for all x∈Mn, or φ(x)=u(x+x0)tv and ψ(x)=u(x−x0)tv for all x∈Mn.
Abstract: Publication date: Available online 3 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Mohammad Reza Oboudi Let G be a simple connected graph with vertices v1,…,vn. The distance matrix of G, denoted by D(G), is the n×n matrix whose (i,j)-entry is equal to d(vi,vj) (length of a shortest path between vi and vj). By the distance spectral radius of G that is denoted by μ(G), we mean the largest eigenvalue of the distance matrix of G. Let X=(m1,…,mt) and Y=(n1,…,nt), where m1≥⋯≥mt≥1 and n1≥⋯≥nt≥1 are integer. We say X majorizes Y and let X⪰MY, if for every j, 1≤j≤t−1, ∑i=1jmi≥∑i=1jni with equality if j=t. In this paper we find a relation between the majorization and the distance spectral radius of complete multipartite graphs. We show that if (m1,…,mt)⪰M(n1,…,nt) and (m1,…,mt)≠(n1,…,nt...
Abstract: Publication date: Available online 3 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Haibin Chen, Liqun Qi, Louis Caccetta, Guanglu Zhou The well-known Birkhoff-von Neumann theorem states that a doubly stochastic matrix is a convex combination of permutation matrices. In this paper, we present a new concept for doubly stochastic tensors and study a generalization of this theorem for doubly stochastic tensors. Particularly, we prove that each permutation tensor is an extreme point of the set of doubly stochastic tensors, and the Birkhoff-von Neumann theorem holds for doubly stochastic tensors. Furthermore, an algorithm is proposed to find a convex combination of permutation tensors for any doubly stochastic tensor.
Abstract: Publication date: Available online 2 September 2019Source: Linear Algebra and its ApplicationsAuthor(s): Diego Aranda-Orna, Alejandra S. Córdova-Martínez We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra C and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of C⊗n and Cn are isomorphic.On the other hand, we prove that the automorphism group schemes of a Smirnov algebra T(C) (a 35-dimensional simple exceptional structurable algebra constructed from a Cayley algebra C) and C are isomorphic. This is used to obtain classifications, up to equivalence and up to isomorphism, of the group gradings on Smirnov algebras.
Abstract: Publication date: Available online 30 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Wai Leong Chooi, Kiam Heong Kwa, Li Yin Tan Let n⩾2 be an integer and let F2 be the Galois field of two elements. Let Tn(F2) be the ring of n×n upper triangular matrices over F2 with unity In. In this paper we characterize commuting additive maps ψ:Tn(F2)→Tn(F2) on invertible matrices, i.e., additive maps ψ satisfying ψ(A)A=Aψ(A) for all invertible matrices A∈Tn(F2). We show that when n⩾4, there exist scalars λ,α,β1,β2∈F2 and matrices H,K∈Tn(F2) and X1,…,Xn∈Tn(F2) satisfying X1+⋯+Xn=0 such thatψ(A)=λA+tr(HtA)In+tr(KtA)E1n+
Abstract: Publication date: Available online 30 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Noe Angelo Caruso, Paolo Novati In this paper we analyze the behavior of the LSQR algorithm for the solution of compact operator equations in Hilbert spaces. We present results concerning existence of Krylov solutions and the rate of convergence in terms of an ℓp sequence where p depends on the summability of the singular values of the operator. Under stronger regularity requirements we also consider the decay of the error. Finally we study the approximation of the dominant singular values of the operator attainable with the bidiagonal matrices generated by the Lanczos bidiagonalization and the arising low rank approximations. Some numerical experiments on classical test problems are presented.
Abstract: Publication date: Available online 29 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): G.A. Bagheri-Bardi, A. Elyaspour, G.H. Esslamzadeh Following a recent work of authors [1], we establish the algebraic analogs of three major decomposition theorems of Nagy-Foias-Langer, Halmos-Wallen and Fishel in Baer ⁎-rings. This approach not only provides purely algebraic and shorter proofs of these decomposition theorems, but also suggests the possibility of eliminating the role of analytic structure in results related to the invariant subspace theory.
Abstract: Publication date: Available online 27 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Yusuke Higuchi, Renato Portugal, Iwao Sato, Etsuo Segawa Staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well-known discrete-time quantum walk models. In this work, we address the class of 2-tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator. By interpreting the evolution operator as a quantum Markov chain on an underlying multigraph, we define the concept of quantum detailed balance, which helps to obtain the eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum Markov chain. To obtain the remaining eigenvectors, we have to use the quantum detailed balance conditions. If the quantum Markov chain has a quantum detailed balance, there is an eigenvector for each fundamental cycle of the underlying multigraph. If the quantum Markov chain does not have a quantum detailed balance, we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors. We exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice (the line graph of the hexagonal lattice), which has symmetry properties that help in the calculation process.
Abstract: Publication date: Available online 23 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Chun-Hua Guo For a square matrix with all eigenvalues in a suitable region in the complex plane, its principal pth root exists and is the limit of the convergent sequence generated by Newton's method or Chebyshev's method, starting from the identity matrix. Such a region is called a convergence region for Newton's method or Chebyshev's method. In this paper, we obtain explicit convergence regions that significantly expand existing ones.
Abstract: Publication date: 1 December 2019Source: Linear Algebra and its Applications, Volume 582Author(s): Wenchang Chu A class of tridiagonal matrices are examined and characterized. Their spectrum, the left and right eigenvectors as well as their scalar products will be determined. The determinants of the two matrices composed by the left and right eigenvectors are also evaluated in closed forms.
Abstract: Publication date: 1 December 2019Source: Linear Algebra and its Applications, Volume 582Author(s): Mark Pankov Let H be a complex Hilbert space and let C be a conjugacy class of rank k self-adjoint operators on H with respect to the action of the group of unitary operators. Under the assumption that dimH≥4k we describe all bijective transformations of C preserving the commutativity in both directions. In particular, it follows from this description that every such transformation is induced by a unitary or anti-unitary operator only in the case when for every operator from C the dimensions of eigenspaces are mutually distinct.
Abstract: Publication date: Available online 20 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Sriparna Chattopadhyay, Kamal Lochan Patra, Binod Kumar Sahoo We study the Laplacian eigenvalues of the zero divisor graph Γ(Zn) of the ring Zn and prove that Γ(Zpt) is Laplacian integral for every prime p and positive integer t≥2. We also prove that the Laplacian spectral radius and the algebraic connectivity of Γ(Zn) for most of the values of n are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of n. The values of n for which algebraic connectivity and vertex connectivity of Γ(Zn) coincide are also characterized.
Abstract: Publication date: Available online 16 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): S. Pilipović, B. Prangoski, M. Žigić We analyse various exponential off-diagonal decay rates of the elements of infinite matrices and their inverses. It is known that such decay of the elements of an infinite matrix does not imply inverse–closeness, i.e. the inverse, if exists, does not have the same order of decay. We discuss some consequences and extensions of this result.
Abstract: Publication date: Available online 16 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Sun Kwang Kim, Han Ju Lee We study the Birkhoff-James orthogonality on operator spaces in terms of the Bhatia-Šemrl property. We first prove that every functional on a Banach space X has the Bhatia-Šemrl property if and only if X is reflexive. We also find some geometric conditions of Banach space which ensure the denseness of operators with Bhatia-Šemrl property. Finally, we investigate operators with the Bhatia-Šemrl property when a domain space is L1[0,1] or C[0,1].
Abstract: Publication date: Available online 16 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): D. Bessades, G. Leal, R. dos Santos, A.C. Vieira Let m be a positive integer and M2m(F) be the algebra of 2m×2m matrices over an algebraically closed field of characteristic zero F endowed with the transpose or the symplectic involution. In this paper, we construct a basis B of M2m(F) over F such that ±B is a group whose elements are symmetric or skew with respect to the given involution. Moreover all elements of this basis commute or anti commute among themselves. The construction is based on a specific irreducible representation of ±B, an extra-special 2-group. As an application, this basis solves the problem on finding the minimal degree of a standard polynomial identity in symmetric variables of (M2m(F),s), where s is the symplectic involution.
Abstract: Publication date: Available online 16 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Carl C. Cowen, William Johnston, Rebecca G. Wahl Given an n×n matrix A over C and an invariant subspace N, a straightforward formula constructs an n×n matrix N that commutes with A and has N=kerN. For Q a matrix putting A into Jordan canonical form, J=Q−1AQ, we get N=Q−1M where M= ker(M) is an invariant subspace for J with M commuting with J. In the formula J=PZT−1Pt, the matrices Z and T are m×m and P is an n×m row selection matrix. If N is a marked subspace, m=n and Z is an n×n block diagonal matrix, and if N is not a marked subspace, then m>n and Z is an m×m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a finite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A.
Abstract: Publication date: Available online 14 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Komla Domelevo, Stefanie Petermichl, Kristina Ana Škreb We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix–valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding. We prove the optimal dependence of the embedding on this quantity. We show that any improvement of a recent matrix weighted bilinear embedding, featuring a scalar Carleson sequence and inner products instead of norms must fail. In particular, replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products. Any formulation using norms, even in the presence of a scalar Carleson sequence must fail. As a positive result, we prove the so–called matrix weighted redundancy condition in full generality.
Abstract: Publication date: Available online 14 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Steven Klee, Matthew T. Stamps The weighted spanning tree enumerator of a graph G with weighted edges is the sum of the products of edge weights over all the spanning trees in G. In the special case that all of the edge weights equal 1, the weighted spanning tree enumerator counts the number of spanning trees in G. The Weighted Matrix-Tree Theorem asserts that the weighted spanning tree enumerator can be calculated from the determinant of a reduced weighted Laplacian matrix of G. That determinant, however, is not always easy to compute. In this paper, we show how two well-known results from linear algebra, the Matrix Determinant Lemma and the method of Schur complements, can be used to elegantly compute the weighted spanning tree enumerator for several families of graphs.
Abstract: Publication date: Available online 9 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Sina Bittens, Gerlind Plonka In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II for reconstructing a real input vector of length N with short support of length m from its discrete cosine transform of type II if an upper bound M on the support length m is known. The resulting algorithm only uses real arithmetic, has a sublinear runtime of O(MlogM+mlog2NM) and requires O(M+mlog2NM) samples. For m and M approaching the vector length, the runtime and sampling requirements approach those of a regular IDCT-II for vectors with full support.
Abstract: Publication date: Available online 9 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Josep Àlvarez Montaner We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this generating function explicitly as a rational function.
Abstract: Publication date: Available online 9 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): John Bamberg, Klaus Metsch We investigate intriguing sets of an association scheme introduced by Penttila and Williford (2011) that was the basis for their construction of primitive cometric association schemes that are not P-polynomial nor the dual of a P-polynomial scheme. In particular, we give examples and characterisation results for the four types of intriguing sets that arise in this scheme.
Abstract: Publication date: Available online 9 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Mohsen Aliabadi, Mano Vikash Janardhanan In this paper, we introduce the notions of matching matrices in groups and vector spaces, which lead to some necessary conditions for existence of acyclic matching in abelian groups and its linear analogue. We also study the linear local matching property in field extensions to find a dimension criterion for linear locally matchable bases. Moreover, we define the weakly locally matchable subspaces and we investigate their relations with matchable subspaces. We provide an upper bound for the dimension of primitive subspaces in a separable field extension. We employ MATLAB coding to investigate the existence of acyclic matchings in finite cyclic groups. Finally, a possible research problem on matchings in n-groups is presented. Our tools in this paper mix combinatorics and linear algebra.
Abstract: Publication date: Available online 8 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): E.I. Milovanović, M.M. Matejić, I.Ž. Milovanović Let G=(V,E) be a simple connected graph of order n and size m, and let ρ1≥ρ2≥⋯≥ρn−1>ρn=0, be the normalized Laplacian eigenvalues of G. The greatest eigenvalue, ρ1, is referred to as the normalized Laplacian spectral radius. The Laplacian incidence energy, the Randić energy, and Kemeny's constant of G, are defined by LIE(G)=∑i=1n−1ρi, RE(G)=∑i=1n ρi−1 , and K(G)=∑i=1n−11ρi, respectively. New lower bounds for the normalized Laplacian spectral radius are obtained. The acquired bounds are then used to obtain new bounds for the LIE(G). In addition, some relations between LIE(G), K(G), RE(G) are determined.
Abstract: Publication date: Available online 7 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Marcin Gąsiorek We present a complete Coxeter spectral classification of finite one-peak posets I=T∪{⁎} that are principal in the sense that their symmetric Gram matrix GI:=12(CI+CItr)∈M I (Q) is positive semi-definite of rank I −1, where CI∈M I (Z) is the incidence matrix of I; equivalently: posets T with the quadratic Tits form qT:Z T +1→Z (in the sense of Drozd) non-negative and the kernel KerqT:={v∈Z T +1;qT(v)=0}⊆Z T +1 infinite cyclic.We continue the study started in M. Gąsiorek, D. Simson (2012) [11]. We show that, up to the strong Gram Z-congruence and the weak Gram Z-congruence, every one-peak principal poset is uniquely determined by its complex Coxeter spectrum speccI⊆C. Moreover, we calculate the exact number of all such posets of a given size (up to isomorphism) and present their Hasse quivers.We finish the paper with conjecture on the exact number of principal (positive) posets. According to our knowledge no formula is known for the number of non-isomorphic posets in a general case.
Abstract: Publication date: Available online 6 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Paul Barry Every Riordan array has what we call a horizontal half and a vertical half. These halves of a Riordan array have been studied separately before. Here, we place them in a common context, showing that one may be obtained from the other. Using them, we provide a canonical factorization of elements of the associated or Lagrange subgroup of the Riordan group. The vertical half matrix is shown to be an element of the hitting-time group. We also ask and answer the question: given a Riordan array, when is it the half (either horizontal of vertical) of a Riordan array'
Abstract: Publication date: Available online 6 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): István Mező, José L. Ramírez In this paper we extend the notion of type B derangements in two ways. First, we refine this notion by studying type B derangements when the number of cycles is fixed. Second, we introduce type B permutations when the number of colors applied to the elements is not necessarily two. We use generating functions, combinatorial arguments and Riordan arrays to study the combinatorial matrices associated to these derangements.
Abstract: Publication date: Available online 6 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Se-Jin Kim, Arthur Mehta Non-commutative graph theory is an operator space generalization of graph theory. Well known graph parameters such as the independence number and Lovász theta function were first generalized to this setting by Duan, Severini, and Winter [1]. In [11], Stahlke introduces a generlazation of the chromatic number of a graph and obtains a Lovász sandwich inequality for certain operator spaces.We introduce two new generalizations of the chromatic number to non-commutative graphs and provide an upper bound on the parameter of Stahlke. We provide a generalization of the graph complement and show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabadussi's Theorem and Hedetniemi's conjecture to non-commutative graphs.
Abstract: Publication date: Available online 5 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Anna Skripka, Maxim Zinchenko We consider Taylor approximations of matrix functions and establish bounds controlling the size of a perturbation in terms of the size of the Taylor remainder or, equivalently, the associated spectral shift function(s). This can be viewed as a stability property for the inverse problem of recovering the perturbation from the spectral shift function(s). From our bounds we derive a uniqueness result for spectral sums: if the trace of the Taylor remainder of order n≥2 equals zero for a finite number of monomials, then the perturbation is zero. This uniqueness result is in contrast with nonuniqueness of the first order Taylor remainder. As an application we characterize equality of two graphs in terms of self-returning walks.
Abstract: Publication date: Available online 5 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Owe Axelsson, Zhao-Zheng Liang A special two-by-two block matrix form arises in many important applications. Extending earlier results it is shown that parameter modified versions of a very efficient preconditioner does not improve its rate of convergence. This holds also for iterative refinement methods corresponding to a few fixed steps of the Chebyshev accelerated method. The parameter version can improve the defect-correction method but the convergence of this method is slower than an iterative refinement method with an optimal parameter. The paper includes also a discussion of how one can save computer elapsed times by avoiding use of global inner products such as by use of a Chebyshev accelerated method instead of a Krylov subspace method. Since accurate and even sharp eigenvalue bounds are available, the Chebyshev iteration method converges as fast as the Krylov subspace method.
Abstract: Publication date: Available online 5 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Mao-Ting Chien, Hiroshi Nakazato, Jie Meng We characterize the diameter and width of the numerical range, present an algorithm for computing the diameter and width of the numerical range, and formulate the diameters of the numerical ranges of unitary bordering matrices for lower dimensions. We also determine the boundaries of the numerical ranges of certain nilpotent Toeplitz matrices to be curves of constant width.
Abstract: Publication date: Available online 5 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Yaroslav Shitov Let P be a preorder relation on a finite set G. The algebra CG×G[G] consists of all complex matrices (with rows and columns indexed by G) which have zeros at those positions (i,j) which are not in P. A subset J⊂G is called P-convex if the conditions a,c∈J, (a,b)∈P, (b,c)∈P imply b∈J. A matrix M∈CG×G[G] is said to satisfy the P-rank/trace conditions ifrankM[J J]⩽traceM[J J]∈Z+ holds for the restriction M[J J] of M to any P-convex set J. A preorder P is called rank/trace complete if any matrix satisfying the P-rank/trace conditions is a sum of rank-one idempotents in CG×G[G]. In this note, we provide an example of a partial order that is not rank/trace complete.
Abstract: Publication date: Available online 5 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Arthur Bik We prove that the inverse limit of the sequence dual to a sequence of Lie algebras is Noetherian up to the action of the direct limit of the corresponding sequence of classical algebraic groups when the sequence of groups consists of diagonal embeddings. We also classify all conjugation-stable closed subsets of the space of N×N matrices.
Abstract: Publication date: Available online 5 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): Fernando Chamizo Bounding the number of morphisms between compact Riemann surfaces is a long standing problem coming from complex and algebraic geometry. We show that linear algebra techniques allow to improve the known results when we assume a kind of condition number bound for the period matrix.
Abstract: Publication date: Available online 1 August 2019Source: Linear Algebra and its ApplicationsAuthor(s): E. Macías-Virgós, M.J. Pereira-Sáez, Daniel Tanré Let Sp(n) be the symplectic group of quaternionic (n×n)-matrices. For any 1≤k≤n, an element A of Sp(n) can be decomposed in A=[αTβP] with P a (k×k)-matrix. In this work, starting from a singular value decomposition of P, we obtain what we call a relative singular value decomposition of A. This feature is well adapted for the study of the quaternionic Stiefel manifold Xn,k, and we apply it to the determination of the Lusternik-Schnirelmann category of Sp(k) in X2k−j,k, for j=0,1,2.
Abstract: Publication date: Available online 31 July 2019Source: Linear Algebra and its ApplicationsAuthor(s): Daochang Zhang, Dijana Mosić, Tin-Yau Tam We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a 2×2 block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the 2×2 block matrix are obtained and some of them generalize several results in the literature.
Abstract: Publication date: Available online 30 July 2019Source: Linear Algebra and its ApplicationsAuthor(s): N. Atreas, N. Karantzas, M. Papadakis, T. Stavropoulos In this paper, we propose a new method for the construction of multi-dimensional, wavelet-like families of affine frames, commonly referred to as framelets, with specific directional characteristics, small and compact support in space, directional vanishing moments (DVM), and axial symmetries or anti-symmetries. The framelets we construct arise from readily available refinable functions. The filters defining these framelets have few non-zero coefficients, custom-selected orientations and can act as finite-difference operators.
Abstract: Publication date: Available online 29 July 2019Source: Linear Algebra and its ApplicationsAuthor(s): Mohamad Badaoui, Alain Bretto, Bassam Mourad Like Cayley graphs, G-graphs are graphs that are constructed from groups but they correspond to alternative constructions. The purpose of this article is to study the connection between these two types of graphs. Such a connection opens up a possible pathway between these two theories and thus investigating certain problems from one of these areas might be easier to tackle when dealt with them as problems in the other. First, we show the existence of a link that connects classes of these two types of graphs, and then we investigate the implications of this result on certain open problems in the theory of Cayley graphs. In particular, we show that computing the spectra of a certain infinite family of Cayley graphs can be easily realized via the use of G-graphs. In the process, general results concerning G-graphs and the spectra of a hypergraph are presented. Finally, we use a certain graph operation to present a new alternative tool for constructing integral graphs.
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