In the field of random walks, the mean first passage time matrix and the Kemeny's constant allow us to deepen into the study of networks. For a transition matrix P, we can observe in the literature how the authors characterize mean first passage time using generalized inverses of I−P and its associated group inverse. In this paper, we focus on obtaining expressions for the mentioned parameters in terms

We answer to a question posed recently in reference [Lovitz B, Petrov F. A generalization of Kruskal's theorem on tensor decomposition. Available at arXiv 2103.15633; 2021], proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition.

A 2-edge-connected graph G is minimally 2-edge-connected if deleting arbitrary edge of G always leaves a graph which is not 2-edge-connected. In this paper, we completely characterize the (minimally) 2-edge-connected graphs with given size having the maximal signless Laplacian (Laplacian) index.

Let Mn be the set of n×n complex matrices, and for ε>0 and A∈Mn, let σε(A) denote the ε−pseudo spectrum of A. Maps Φ on Mn which preserve the skew Lie product of matrices in a sense that σε(Φ(A)Φ(B)−Φ(B)Φ(A)∗)=σε(AB−BA∗)(A,B∈Mn)are characterized, with no surjectivity assumption on them. Analogous description for the skew product on matrices is also noted.

Let p1

In this paper, we introduce new quantum divergences of the form Φ(A,B)=Tr(AσB−AτB),A,B>0,where σ and τ are Kubo–Ando operator means such that σ≥τ. More precisely, we show that Φ(A,B) is a quantum divergence when σ is the weighted Kubo–Ando matrix power mean and τ is the weighted geometric mean. In addition, we construct a new quantum Hellinger-type divergence using the linear approximation of the function

We establish a trace inequality of symplectic matrices via a more general trace minimization theorem. As a consequence, we derive another proof of a result in [R. Bhatia, T. Jain, On symplectic eigenvalues of positive-definite matrices, J. Math. Phys., 56 (2015),112201.].

For every n∈N and every field K, let A(n,K) be the vector space of the antisymmetric (n×n)-matrices over K. We say that an affine subspace S of A(n,K) has constant rank r if every matrix of S has rank r. Define AantisymK(n;r)={S|SaffinesubspaceofA(n,K)ofconstantrankr},aantisymK(n;r)=max{dim⁡S∣S∈AantisymK(n;r)}.In this paper, we prove the following formulas: for n≥2r+2 aantisymR(n;2r)=(n−r−1)r;for n = 2r

We present a complete characterization of the right-symmetric points in the one sum of two Banach spaces. We also obtain some basic properties of the left-symmetric (right-symmetric) points in the p sum, 1≤p≠2<∞ ( 1

We consider a face of the polytope of doubly stochastic matrices, whose non-zero entries coincide with that of Vl,m,n=[0l,l0l,mJl,n0m,lImJm,nJn,lJn,mJn,n]. Here, 0r,s is the r×s zero matrix, Ju,v denotes the u×v matrix all of whose entries are 1 and Im is the identity matrix of order m. We determine the minimum permanent and minimizing matrices on this face of the polytope of doubly stochastic matrices

We introduce and study m-heavy subspaces of a Banach space and obtain a complete analytic characterization of the same. The importance of m-heavy subspaces in studying the geometry of a Banach space is illustrated by establishing its various applications, including a strengthening of the classical Birkhoff–James orthogonality, duality theory and linear Hahn–Banach extension operators.

Let G be a connected graph with adjacency matrix A(G) and distance matrix D(G). The adjacency-distance matrix of G is defined as S(G)=D(G)+A(G). In this paper, S(G) is generalized by the convex linear combinations Sα(G)=αD(G)+(1−α)A(G)where α∈[0,1]. Let ρ(Sα(G)) be the spectral radius of Sα(G). This paper presents results on Sα(G) with emphasis on ρ(Sα(G)) and some results on S(G) are extended to all

The stable index of a 0–1 matrix A is defined to be the smallest integer k such that Ak+1 is not a 0–1 matrix if such an integer exists; otherwise the stable index of A is defined to be infinity. We characterize the set of stable indices of 0–1 matrices with a given order.

Let n≥ 2 be a natural number, and denote by Mn the algebra of all n×n matrices over an algebraically closed field F of zero characteristic. Let also K1 and K2 be two non-empty proper subsets of F. In this paper, we characterize linear maps φ on Mn having the property that, for every T∈Mn, the spectrum of T is a subset of K1 if and only if the spectrum of φ(T) is a subset of K2. We obtain a similar

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator T∗T of a

New multiplicative perturbation bounds for orthogonal projection in a general unitarily invariant norm, the Q-norm and the spectral norm are derived. These bounds always improve the existing bounds. An example is given to show that these new bounds are optimal.

For a non-empty set I, the sub-defect of an I×I doubly substochastic matrix A=[aij]i,j∈I, denoted by sd(A), is the smallest cardinal number α for which there is a set J with card(J)=α, I∩J=∅, and there exists a doubly stochastic matrix D=[dij]i,j∈I∪J which contains A as a sub-matrix. In this paper, we show the set of all finite sub-defect matrices is closed under multiplication. We also show that the

Let Bn,k,a be the set of weighted k-uniform connected bicyclic hypergraphs of order n with positive integer weights and fixed total weight sum a, where n, k and a are positive integers with k≥4 and a≥n+1k−1≥3. An α-normal labelling method is proposed for comparing the spectral radii of the weighted hypergraphs under consideration. The weighted hypergraph having the maximum spectral radius among the

Nekrasov matrices play an important role in various scientific disciplines. The estimation of infinity norm bounds for the inverse of Nekrasov matrices brings a lot of convinces in many fields. In this paper, we introduce two new bounds for the inverse of Nekrasov matrices. The advantages of our bounds and numerical examples are also presented.

For a simple graph G on n vertices the Seidel Estrada index of G, denoted by SEE(G), is defined as SEE(G)=∑i=1neθi, where θ1,…,θn are the Seidel eigenvalues (the eigenvalues of the Seidel matrix) of G. In this paper, we find the maximum and minimum values of the Seidel Estrada index among all graphs with the fixed number of vertices. Our results confirm some conjectures on Seidel Estrada index of graphs

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is (g,gq,gq2,…,gqn−1), where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse

For positive integers m1,m2,…,mh, a generalized balanced tree T(m1,m2,…,mh) is a rooted tree of height h such that every vertex of depth i has mi+1 children, 0≤i≤h−1. The distance matrix D(G) of a simple connected graph G of order n is an n×n matrix whose (i,j)th entry is the distance between ith and jth vertices. A connected graph G is called a k-partitioned transmission regular graph if there exists

Several spectral radii formulas for infinite bounded non-negative matrices in max algebra are obtained. We also prove some Perron–Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are similar to results on Frobenius normal form of n×n matrices. Some continuity results are also established.

Suppose that m,k,t are integers with m,k≥ 1 and 0≤t and A is the k×k matrix with the (i,j)-entry (t+(j−1)mi−1). When t = 0 and m = 1, this is the upper triangular Pascal matrix. Here, first we study the properties of this matrix, in particular, we find its determinant and its LDU decomposition and also study its inverse. Then by using this matrix we present a generalization of the Mattson-Solomon transform

We generalize the theory of homotopic deviation of square complex matrices to regular matrix pencils. To this end, we study the existence and the analyticity of the resolvent of the matrix pencils whose matrices are under homotopic deviation with the deviation parameter t∈C. Moreover, we investigate and identify the limits of both the resolvent and the spectrum of the deviated matrix pencils, as |t|→∞

We compute the numerical index of the two-dimensional real Lp space for 65⩽p⩽1+α0 and α1⩽p⩽6, where α0 is the root of f(x)=1+x−2−(x−1x+x1x) and 11+α0+1α1=1. This, together with the previous results in Merí and Quero [On the numerical index of absolute symmetric norms on the plane. Linear Multilinear Algebra. 2021;69(5):971–979] and Monika and Zheng [The numerical index of ℓp2. Linear Multilinear Algebra

We show that the tempered spectrum of a sufficiently small perturbation of a sequence of matrices varies little, in the sense that it is contained in a small open neighbourhood of the tempered spectrum of the original sequence. In addition, we show that for perturbations that decay exponentially, all the Lyapunov exponents of the perturbation belong to the tempered spectrum of the original sequence

The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes

This paper is an attempt to give an axiomatic approach to the investigation of various kinds of generalizations of Drazin invertibility in Banach algebras. We shall say that an element a of a Banach algebra A is generalized Drazin invertible relative to a regularity R if there is b∈A such that ab=ba, bab=b and σR(a−aba)⊂{0}. The concept of Koliha-Drazin invertible elements, as well as some generalizations

In this paper, a new preconditioned AOR method for solving the multi-linear systems with M-tensor is presented. The convergence of the new method and the corresponding comparison for the spectral radius of preconditioned iterative tensors are given. Numerical examples demonstrate that the new method is feasible and effective. As an application, we also show the efficiency of our preconditioner for

Vertices in the graph of a square matrix over a field may be classified as to how their removal changes the geometric multiplicity of an identified eigenvalue. There are three possibilities: +1 (Parter); no change (neutral); and −1 (downer). When the graph is a tree, the ‘downer branch mechanism’ distinguishes the Parter vertices. Here, we discover how this mechanism generalizes for general graphs

Let G be a simple graph with the adjacency matrix A(G). Let τ(G) denote the smallest positive eigenvalue of A(G). In 1990, Pavlíková and Kr c˘-Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let

We investigate the converse of the known fact that if the Gershgorin discs of a real n-by-n matrix may be separated by positive diagonal similarity, then the eigenvalues are real. In the 2-by-2 case, with appropriate signs for the off-diagonal entries, we find that the converse is correct, which raises several questions. First, in the 3-by-3 case, the converse is not generally correct, but, empirically

The present paper is devoted to the study of 2-local derivations on the Schrödinger-Virasoro Algebra, which is an infinite-dimensional Lie algebra with three outer derivations. We prove that all 2-local derivations on the Schrödinger-Virasoro Algebra are derivations.

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general

In this paper, we discuss and present new sharp inequalities for the numerical radii of Hilbert space operators. In particular, if A and B are bounded linear operators on a Hilbert space, we present new upper bounds for ω(A∗B). The main tool to obtain our results is using block matrix techniques. Among many interesting results, and as an application of the new inequalities, we obtain the following

In this paper, we propose a sparse least squares (SLS) optimization model for solving multilinear equations, in which the sparsity constraint on the solutions can effectively reduce storage and computation costs. By employing variational properties of the sparsity set, along with differentiation properties of the objective function in the SLS model, the first-order optimality conditions are analysed

In this paper, we give spectral radius inequalities for matrices with entries from a Banach algebra. An emphasis will be given to matrices with commuting entries. Our inequalities are natural generalizations of earlier known inequalities for operator matrices.

This note is a corrigendum to our recent paper ‘The periodic determinantal property for (0,1) double banded matrices’. For the sake of clarity and completeness, we provide several additional comments and examples.

In this paper, we are interested in studying Bishop–Phelps–Bollobás type properties related to the denseness of the operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and the second numerical index strictly positive satisfies the Bishop–Phelps–Bollobás point property, and we see that the one-dimensional space is the only one with both the numerical

This paper is concerned with the iterative solutions to a class of generalized Sylvester quaternion tensor equations. From the Kronecker product and the properties of quaternion tensors, we propose the biconjugate A-orthogonal residual stabilized and the generalized product-type biconjugate A-orthogonal residual methods in their tensor forms for solving the original tensor equation. As an application

In this paper, further characterizations of selfadjoint operators on Hilbert spaces are established. Thus the recent results concerning two-sided removal and cancellation laws associated with a Hermitian matrix (Central University of Finance and Economics, DOI: https://doi.org/10.21203/rs.3.rs-1104901/v1) are extended to the infinite dimensional setting with proofs based on operator matrices.

In this paper, we give a complete classification of all free U(CL0⊕CY0⊕CM0)-modules of rank 1 over a Schrödinger-Virasoro type algebra tsv.

We find the necessary and sufficient conditions for the multiplication operator MV:c0(X)→lp(Y) defined by MV((xn)n∈N):=(Vn(xn))n∈N to be 2-summing. Some examples are given.

The Quotient Lifting Property (QLP) for pairs of spaces and its relation to the proximinality of subspaces are studied. Even though this property is not hereditary, for finite-dimensional subspaces of L1(μ), we show that for a choice of a basis, all subspaces spanned by any sub-collection from the base, generate subspaces with QLP. Spaces of dimension greater than two with the total quotient lifting

Due to the redundancy property of frames, the stable decomposition of a vector in the separable Hilbert space H allows the flexibility of choosing different types of duals for a frame. For a second countable locally compact group G (not necessarily abelian) and a closed abelian subgroup Γ, we study the properties of oblique Γ-translation generated (Γ-TG) duals for a continuous frame in L2(G). Two types

The concept of CπR-matrix is extended to CπR-tensor, which also generalizes the concept of C-tensor. A necessary and sufficient condition for a tensor to be a CπR-tensor is provided. We analyse decompositions of CπR-tensors and prove that CπR-tensors are nonsingular. Positive linear combinations and Hadamard product of two CπR-tensors are also discussed. Finally, some properties of BπR-tensor are given

Let A and B be n×n positive semidefinite matrices, f:[0,∞)→[0,∞) be an increasing convex function with f(0)=0, and let Φ:Mn(C)→Mn(C) be a positive unital linear map. It is shown that ∏j=1ksj(f4(Φ(A+B2)))≤2−4∏j=1k(sj(Φ2(f2(A))+Φ(f2(A)))+‖Φ(f2(B))‖+1)×(sj(Φ2(f2(B))+Φ(f2(B)))+‖Φ(f2(A))‖+1)for k=1,2,…,n, where sj(T) and ‖T‖ are the jth singular value and the spectral norm of T, respectively. Applications

For real matrices selfadjoint in an indefinite inner product there are two special canonical Jordan forms, that is (i) flipped orthogonal (FO) and (ii) γ-conjugate symmetric (CS). These are the classical Jordan forms with certain additional properties induced by the fact that they are H-selfadjoint. In this paper, we prove that for any real H-selfadjoint matrix, there is a γ-FOCS Jordan form that is

We give real representations of An (or etA) based on AnPμ for a real square matrix A, where Pμ is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares

Let Φ,Ψ be symmetrically norming (s.n.) functions, let CΨ(H) be the ideal of compact Hilbert space operators, associated with the s.n. function Ψ, p⩾2 and let A,B,X∈B(H) be such that A, B are accretive and AX+XB∈CΨ(H). Then A∗+AXB+B∗∈CΨ(H) as well, and ||A∗+AXB+B∗||Ψ⩽||AX+XB||Ψ,under any of the following conditions: Ψ:=Φ(p)∗ and A (resp. B∗) is quasinormal operator with its adjoint operator being 2-hyperaccretive

In this paper, we introduce the limit, unique solution of the nonlinear equations, geodesic property, tolerance relations and pinch on the spectral geometric mean for two positive definite operators. We show that the spectral geometric mean is a geodesic with respect to some semi-metric. We also prove that the tolerance relation on determinant one matrices can be characterized by the spectral geometric

The weight of a partition is the sum of its nonzero coordinates. Let R and S be partitions of the same weight and A(R,S) be the class of all (0,1)-matrices with row sums R and column sums S. For a positive integer t, the t-term rank of a matrix A∈A(R,S), denoted ρt(A), is defined as the largest number of 1's in A with at most one 1 in each column and at most t 1's in each row. The term rank partition

ABSTRACT Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field Fq of length mℓ for every positive even integer ℓ. In this paper, we study the case where xm−1 has an arbitrary number of irreducible factors in Fq[x]; in the previous studies, only some special cases where xm−1 has exactly two or three irreducible factors in Fq[x], were studied. Firstly

The main goal of this article is to present new inequalities for the spectral geometric mean A♮tB of two positive definite operators A, B on a Hilbert space. The obtained results complement many known inequalities for the geometric mean A♯tB. In particular, explicit comparisons between A♮tB and A♯tB are given, with applications towards Ando-type inequalities and Ando-Hiai inequalities for A♮tB and

Some new sufficient conditions are given, which ensure that the k-subdirect sum of Nekrasov matrices and strictly diagonally dominant matrices is in the class of Nekrasov matrices. In addition, several examples are also given to illustrate the conditions presented.

In this paper, we prove several conjectures raised by Zhi-Wei Sun on determinants and permanents by the eigenvectors-eigenvalues identity recently highlighted by Denton, Parke, Tao and Zhang.

In this paper, we study that the algebraic multiplicity of the zero Laplacian eigenvalue of a connected uniform hypergraph. We give the algebraic multiplicity of the zero Laplacian eigenvalue of a hyperstar. For a loose hyperpath, we characterize the algebraic multiplicity of the zero Laplacian eigenvalue by the multiplicities of points in the affine variety defined by the Laplacian eigenvalue equations

Let m, n be integers such that 1