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Let G be a connected graph with adjacency matrix $ A(G) $ A (G) and distance matrix $ \mathcal {D}(G) $ D (G). The adjacency-distance matrix of G is defined as $ S(G) = \mathcal {D}(G) + A(G) $ S (G) = D (G) + A (G). In this paper, $ S(G) $ S (G) is generalized by the convex linear combinations \[ S_{\alpha}(G)=\alpha \mathcal{D}(G)+(1-\alpha)A(G) \] S α (G) = α D (G) + (1 − α) A (G) where $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ]. Let $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) be the spectral radius of $ S_{\alpha }(G) $ S α (G). This paper presents results on $ S_{\alpha }(G) $ S α (G) with emphasis on $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) and some results on $ S(G) $ S (G) are extended to all α in some subintervals of $ [0,1] $ [ 0 , 1 ]. For $ \alpha \in [1/2,1] $ α ∈ [ 1 / 2 , 1 ] , the trees attaining the largest and the smallest $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) among trees of fixed order are determined and it is proved that $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) is a branching index. Moreover, for $ \alpha \in (1/2,1] $ α ∈ (1 / 2 , 1 ] , the graphs that uniquely minimize $ \rho (S_{\alpha }(G)) $ ρ (S α (G)) : among all connected graphs of fixed order and fixed connectivity, and among all connected graphs of fixed order and fixed chromatic number are characterized. [ABSTRACT FROM AUTHOR]

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 index 1 and the Bishop–Phelps–Bollobás point property. We also consider two weaker properties L $ {}_{p,p} $ p , p -nu and L $ {}_{o,o} $ o , o -nu, the local versions of Bishop–Phelps–Bollobás point and operator properties respectively, where the η which appears in their definition does not depend just on $ \varepsilon > 0 $ ε > 0 but also on a state $ (x, x^*) $ (x , x ∗) or on a numerical radius one operator T. We address the relation between the L $ {}_{p,p} $ p , p -nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and $ c_0 $ c 0 are examples of Banach spaces satisfying the L $ {}_{p,p} $ p , p -nu, and we exhibit an example of a Banach space with a strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the L $ {}_{o,o} $ o , o -nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality. [ABSTRACT FROM AUTHOR]

Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for every positive even integer ℓ. In this paper, we study the case where $ x^m-1 $ x m − 1 has an arbitrary number of irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] ; in the previous studies, only some special cases where $ x^m-1 $ x m − 1 has exactly two or three irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for any even positive integer ℓ, where we require that $ q \equiv 1 \pmod {4} $ q ≡ 1 (mod 4) if the index $ \ell \ge 6 $ ℓ ≥ 6. By implementation of our method, we obtain a new optimal binary self-dual code $ [172, 86, 24] $ [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4. [ABSTRACT FROM AUTHOR]

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by $ W(G) $ W (G) , is $ [e,Ae,\ldots,A^{n-1}e] $ [ e , Ae , ... , A n − 1 e ] , where e is the all-ones vector. Let $ G\circ P_m $ G ∘ P m be the rooted product of G and a rooted path $ P_m $ P m (taking an endvertex as the root), i.e. $ G\circ P_m $ G ∘ P m is a graph obtained from G and n copies of $ P_m $ P m by identifying each vertex of G with an endvertex of a copy of $ P_m $ P m . Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and $ m\in \{3,4\} $ m ∈ { 3 , 4 } , respectively \[ \det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m, \] det W (G ∘ P m) = ± a 0 ⌊ m 2 ⌋ (det W (G)) m , where $ a_0 $ a 0 is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any $ m\ge 2 $ m ≥ 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with left star, star, and core partial orders for complex matrices. For each partial order, we present an order-isomorphism between the down-set of a fixed matrix B and a certain set (depending on the partial order) of orthogonal projectors whose matrix sizes can be considerably smaller than that of the matrix B. We study the lattice structure and we give properties of the down-sets. We prove that the down-set of B ordered by the core partial order and by the star partial order are sublattices of the down-set ordered by the left star partial order. We analize the existence of supremum and infimum of two given matrices and we give characterizations of these operations (whenever they exist). Some of the results given in the paper are already known in the literature but we present a different proof based on the previously established order-isomorphism. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let $ n \geq ~2 $ n ≥ 2 be a natural number, and denote by $ \mathcal {M}_{n} $ M n the algebra of all $ n \times n $ n × n matrices over an algebraically closed field $ \mathbb {F} $ F of zero characteristic. Let also $ K_{1} $ K 1 and $ K_{2} $ K 2 be two non-empty proper subsets of $ \mathbb {F} $ F . In this paper, we characterize linear maps φ on $ \mathcal {M}_{n} $ M n having the property that, for every $ T \in \mathcal {M}_{n} $ T ∈ M n , the spectrum of T is a subset of $ K_1 $ K 1 if and only if the spectrum of $ \varphi (T) $ φ (T) is a subset of $ K_2 $ K 2 . We obtain a similar caracterization for the case when $ K_{1} $ K 1 and $ K_{2} $ K 2 have both at most n elements, working with the equality of the spectrum to the fixed subsets instead of the inclusion. [ABSTRACT FROM AUTHOR]

For a simple graph G on n vertices the Seidel Estrada index of G, denoted by $ SEE(G) $ SEE (G) , is defined as $ SEE(G)=\sum _{i=1}^ne^{\theta _i} $ SEE (G) = ∑ i = 1 n e θ i , where $ \theta _1,\ldots,\theta _n $ θ 1 , ... , θ n 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 that have been posed in [M. Hakimi-Nezhaad, M. Ghorbani, On the Estrada Index of Seidel Matrix, Mathematics Interdisciplinary Research5 (2020) 43–54]. [ABSTRACT FROM AUTHOR]

For a non-empty set I, the sub-defect of an $ I\times I $ I × I doubly substochastic matrix $ A=[a_{ij}]_{i,j \in I}, $ A = [ a ij ] i , j ∈ I , denoted by $ \mathrm {sd}(A), $ sd (A) , is the smallest cardinal number α for which there is a set J with $ \mathrm {card}(J) =\alpha, $ card (J) = α , $ I\cap J =\emptyset, $ I ∩ J = ∅ , and there exists a doubly stochastic matrix $ D=[d_{ij}]_{i,j\in I\cup J} $ D = [ d ij ] 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 inequality $ \max \{\mathrm {sd}(A), \mathrm {sd}(B)\} \leq \mathrm {sd}(AB) \leq \min \{n, \mathrm {sd}(A) + \mathrm {sd}(B)\} $ max { sd (A) , sd (B) } ≤ sd (AB) ≤ min { n , sd (A) + sd (B) } which is obtained by Lei Cao et al. remains valid for all finite sub-defect matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 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 of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

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 $ \mathcal {A} $ A is generalized Drazin invertible relative to a regularity $ \mathcal {R} $ R if there is $ b\in \mathcal {A} $ b ∈ A such that $ ab=ba,\ bab=b $ ab = ba , bab = b and $ \sigma _{\mathcal {R}}(a-aba)\subset \{0\} $ σ R (a − aba) ⊂ { 0 }. The concept of Koliha-Drazin invertible elements, as well as some generalizations of this concept are described via the concept of generalized Drazin invertible elements relative to a regularity $ \mathcal {R} $ R which satisfies two properties: (D1) if $ a, b\in \mathcal {R} $ a , b ∈ R , p is an idempotent commuting with a and b, then $ ap+b(1-p)\in \mathcal {R} $ ap + b (1 − p) ∈ R ; (D2) if $ a\in \mathcal {R} $ a ∈ R , then a is almost invertible. If a regularity $ \mathcal {R} $ R satisfies the properties (D1) and (D2), we prove that $ a\in \mathcal {A} $ a ∈ A is generalized Drazin invertible relative to $ \mathcal {R} $ R if and only if 0 is not an accumulation point of $ \sigma _{\mathcal {R}}(a) $ σ R (a). In particular we define and characterize generalized Drazin-T-Riesz invertible elements relative to an arbitrary (not necessarily bounded) Banach algebra homomorphism T and so extend the concept of generalized Drazin-Riesz invertible operators introduced in [Živković-Zlatanović SČ, Cvetković MD. Generalized Kato-Riesz decomposition and generalized Drazin-Riesz invertible operators. Linear Multilinear A. 2017;65(6):1171–1193]. Also we consider generalized Drazin invertibles relative to $ \mathcal {R} $ R in the case when $ \mathcal {R} $ R is the set of Drazin invertibles, as well as when $ \mathcal {R} $ R is the set of Koliha-Drazin invertibles. [ABSTRACT FROM AUTHOR]

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 have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than $ ({q}/{2})+1 $ (q / 2) + 1. [ABSTRACT FROM AUTHOR]

In this paper, a new preconditioned AOR method for solving the multi-linear systems with $ \mathcal {M} $ 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 solving higher order Markov chains. [ABSTRACT FROM AUTHOR]

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, the proposed methods are tested with some colour video restoration problems. Numerical results show the effectiveness of our methods compared with some existing methods. [ABSTRACT FROM AUTHOR]

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 terms of the stationary points. Based on the equivalent characterization of the stationary points, we propose the Newton Hard-Threshold Pursuit (NHTP) algorithm and establish its locally quadratic convergence under some regularity conditions. Numerical experiments conducted on simulated datasets including cases of Completely Positive(CP)-tensors and symmetric strong M-tensors illustrate the efficiency of our proposed NHTP method. [ABSTRACT FROM AUTHOR]

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, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of all free $ U(\mathbb {C}L_0 \oplus \mathbb {C}Y_0\oplus \mathbb {C}M_0) $ U (C L 0 ⊕ C Y 0 ⊕ C M 0) -modules of rank 1 over a Schrödinger-Virasoro type algebra $ \mathfrak {tsv} $ tsv . [ABSTRACT FROM AUTHOR]

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

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. [ABSTRACT FROM AUTHOR]

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 decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained. [ABSTRACT FROM AUTHOR]

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 simultaneously flipped orthogonal and γ-conjugate symmetric. [ABSTRACT FROM AUTHOR]

Let m, n be integers such that 1

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. We compute the algebraic multiplicities of the zero Laplacian eigenvalue of a loose hyperpath. We also show that the algebraic multiplicity of the zero Laplacian eigenvalue is not smaller than the number of irreducible components of the eigenvariety associated with the zero Laplacian eigenvalue for a loose hyperpath. [ABSTRACT FROM AUTHOR]

The weight of a partition is the sum of its nonzero coordinates. Let R and S be partitions of the same weight and $ {\mathcal {A}}(R,S) $ A (R , S) be the class of all $ (0,1) $ (0 , 1) -matrices with row sums R and column sums S. For a positive integer t, the t-term rank of a matrix $ A\in {\mathcal {A}}(R,S) $ A ∈ A (R , S) , denoted $ \rho _t(A) $ ρ t (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 of A is the partition $$\begin{align*} term\ \rho(A) & :=(\rho_1(A)-\rho_0(A),\rho_2(A) \\ & \quad - \rho_1(A),\ldots ,\rho_t(A)-\rho_{t-1}(A),\ldots), \end{align*} $$ term ρ (A) := (ρ 1 (A) − ρ 0 (A) , ρ 2 (A) − ρ 1 (A) , ... , ρ t (A) − ρ t − 1 (A) , ...) , where $ \rho _0(A)=0 $ ρ 0 (A) = 0. Let θ be a partition of weight equal to the number of nonzero coordinates of S. In this paper, we study conditions for the existence of a matrix $ C\in {\mathcal {A}}(R,S) $ C ∈ A (R , S) with $ term\ \rho (C)=\theta. $ term ρ (C) = θ. [ABSTRACT FROM AUTHOR]

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 mean. Moreover, two positive tuples can be pinched by the spectral geometric mean. [ABSTRACT FROM AUTHOR]

In this paper, left Atkinson linear relations defined in a Banach space are considered. The behaviour of such notion in polynomials is studied and a spectral mapping theorem for the corresponding spectrum is deduced. Let $ \boldsymbol {\mathcal {R}} $ R denote any of the following classes: left (right) Atkinson, Fredholm, left (right) Weyl, Weyl, left (right) Browder, Browder linear relations. For a closed and everywhere defined linear relation $ \mathcal {T}, $ T , we prove that under suitable conditions, $ \mathcal {T} = \mathcal {T}_{\mathcal {M}} \oplus \mathcal {T}_{\mathcal {N}}, $ T = T M ⊕ T N , where $ \mathcal {T}_{\mathcal {M}}\in \boldsymbol {\mathcal {R}} $ T M ∈ R and $ \mathcal {T}_{\mathcal {N}} $ T N is a bounded nilpotent operator if and only if $ \mathcal {T} $ T admits a Kato-decomposition and 0 is not an interior point of the $ \boldsymbol {\mathcal {R}} $ R spectrum of $ \mathcal {T}, $ T , $ \sigma _{\boldsymbol {\mathcal {R}}}(\mathcal {T}). $ σ R (T). As an application, some connections results between $ \sigma _{\boldsymbol {\mathcal {R}}}(\mathcal {T}) $ σ R (T) and $ \sigma _{\boldsymbol {\mathcal {DK}}}(\mathcal {T}) $ σ DK (T) are established, where $ \sigma _{\boldsymbol {\mathcal {DK}}}(\mathcal {T}) $ σ DK (T) denotes the Kato-decomposition spectrum of $ \mathcal {T}. $ T. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the q-numerical radius $ \omega _{q}(.) $ ω q (.) of bounded linear operators on Hilbert spaces. More precisely, first, we show that $ \omega _{q}(.) $ ω q (.) defines a norm which is equivalent to the operator norm. Next, the following compatible generalization of Kittaneh's inequality $$\begin{align*} \frac{1}{4}\left(\frac{q}{2-q^{2}}\right)^{2}\parallel T^{*}T +TT^{*}\parallel & \leq \omega_{q}^{2}(T) \leq \frac{(q+2\sqrt{1-q^2})^2}{2}\\ & \quad\times \Vert T^{*}T+TT^{*}\Vert. \end{align*}$$ 1 4 (q 2 − q 2 ) 2 ∥ T ∗ T + T T ∗ ∥ ≤ ω q 2 (T) ≤ (q + 2 1 − q 2 ) 2 2 × ‖ T ∗ T + T T ∗ ‖. is obtained. Finally, some generalizations of q-numerical radius inequalities for composition of operators are established. [ABSTRACT FROM AUTHOR]

In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed–Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed–Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space. [ABSTRACT FROM AUTHOR]

Rayleigh–Ritz projection is one of the efficient procedures to project a large-scale quadratic eigenvalue problem (QEP) into a small-scale QEP by properly choosing a low-dimensional subspace. One common way for solving the projected QEP is to recast it via linearization. In this paper, we establish bounds for backward errors of approximate eigenpairs of QEP relative to those of a linearization. These bounds give useful information to predict the numerical stability of Rayleigh–Ritz projection eigensolvers followed by a linearization. We present results of numerical experiments that support the predictions of the backward error analysis. [ABSTRACT FROM AUTHOR]

In this paper, exponential stability of continuous-time and discrete-time switched $ k\times k $ k × k block triangular systems under arbitrary switching is studied. Firstly, under the assumption that all subsystem matrices are Hurwitz and a family of those corresponding block diagonal matrices is commutative, we prove that a continuous-time switched linear system is exponentially stable under arbitrary switching. Next, under the assumption that all subsystem matrices are Hurwitz and all those block diagonal matrices are normal, it is shown that the same switched system is exponentially stable under arbitrary switching. Further, under similar conditions we prove that a discrete-time switched linear system is exponentially stable under arbitrary switching. After that, illustrative numerical examples of the obtained results are also given. Finally, we prove that $ 3\times 3 $ 3 × 3 normal matrices have nine parameter representations which are useful for numerical examples (in the Appendix). [ABSTRACT FROM AUTHOR]

Let $ \mathcal {L}(\mathscr {H}) $ L (H) denote the $ C^* $ C ∗ -algebra of adjointable operators on a Hilbert $ C^* $ C ∗ -module $ \mathscr {H} $ H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in $ \mathscr {L}(\mathscr {H}) $ L (H). More precisely, an operator $ A\in \mathscr {L}(\mathscr {H}) $ A ∈ L (H) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists $ \nu \in (0, 1) $ ν ∈ (0 , 1) such that \[ \|\langle Ax, y\rangle\|\leq (\|Ax\|\|y\|)^{\nu}(\|Ay\|\|x\|)^{1 - \nu} \quad (x, y \in \mathscr{H}). \] ‖ ⟨ A x , y ⟩ ‖ ≤ (‖ A x ‖ ‖ y ‖) ν (‖ A y ‖ ‖ x ‖) 1 − ν (x , y ∈ H). We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if $ \left \|\langle Ax, y\rangle \right \| \leq \left \|{|A|}^{1/2}x\right \|\left \|{|A|}^{1/2}y\right \| $ ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all $ x, y \in \mathscr {H} $ x , y ∈ H . [ABSTRACT FROM AUTHOR]

In this paper, we study the necessary and sufficient conditions to have a solution pair $ (X, Y) $ (X , Y) to the generalized Sylvester equation AX−Y B = C where $ A, B, C\in \mathcal {L}(\mathcal H) $ A , B , C ∈ L (H) are given. Moreover, we give some connections the generalized Fuglede-Putnam property and the solution pair $ (X, Y) $ (X , Y) of the operator equation AX−Y B = C. Finally, we consider some properties of the operator equation AX−Y B = C when A and B belong to the special classes of operators. [ABSTRACT FROM AUTHOR]

The present paper is devoted to studying local derivations on the Lie algebra $ W(2,2) $ W (2 , 2) which has some outer derivations. Using some linear algebra methods in [1] and a key construction for $ W(2,2) $ W (2 , 2) , we prove that every local derivation on $ W(2, 2) $ W (2 , 2) is a derivation. As an application, we determine all local derivations on the deformed $ \mathfrak {bms}_3 $ b m s 3 algebra. [ABSTRACT FROM AUTHOR]

In the setting of adjointable operators on Hilbert $ C^* $ C ∗ -modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, a full characterization is made on the polar decomposition of the product of three operators. Some applications of this characterization are also dealt with. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-step Noda iteration for computing the Perron root and Perron vector of an irreducible nonnegative matrix by successively solving two linear systems with the same coefficient matrix. For every positive initial vector, the two-step Noda iteration always converges and has a cubic asymptotic convergence rate. As an application, the two-step Noda iteration is applied to compute the smallest eigenpair of irreducible nonsingular M-matrices. Numerical examples are provided for testing the proposed method. [ABSTRACT FROM AUTHOR]

Let Γ be a finite simple graph with adjacency matrix A. Γ is said to be singular if and only if A has an eigenvalue 0. The nullity of Γ, denoted by null $ (\Gamma) $ (Γ) , is the multiplicity of the eigenvalue 0 in the spectrum of Γ. In this paper, we completely determine the singularity of graphs Γ for which the dicylic or semi-dihedral group acts transitively and faithfully on V as a group of automorphisms. Moreover, we give formulas to calculate the nullities for such graphs. [ABSTRACT FROM AUTHOR]

The dangling nodes are the nodes with no out-links in the web graph. It saves many computational costs if the dangling nodes are lumped into one node. In this paper, motivated by so many dangling nodes in web graphs, we mainly develop theoretical results for the modified hyperlink-induced topic search (MHITS) model by the lumping method, and the results are also suitable for the HITS model. There are three main findings in the theoretical analysis. First, MHITS can be lumped, although the matrix involved is not stochastic. Second, the hub vector of the nondangling nodes can be computed separately from dangling nodes, and the hub vector of the dangling nodes is parallel to a vector of all ones in MHITS. Third, the authority vector of the nondangling nodes is difficult to compute separately from dangling nodes. The numerical results not only show the feasibility and effectiveness of theoretical analyses, but also demonstrate that the lumped MHITS method can produce a better initial authority vector after the iteration of computing the hub vector. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Jacobi-type algorithm for the approximate diagonalization of tensors of order $ d\geq 3 $ d ≥ 3 via tensor trace maximization. For a general tensor, this is an alternating least squares algorithm and the rotation matrices are chosen in each mode one-by-one to maximize the tensor trace. On the other hand, for symmetric tensors, we discuss a structure-preserving variant of this algorithm where in each iteration the same rotation is applied in all modes. We show that both versions of the algorithm converge to the stationary points of the corresponding objective functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose the horizontal implicit complementarity problems, which cover many complementarity problems that have wide applications. After reformulating the horizontal implicit complementarity problem as an implicit fixed-point equation, the modulus-based matrix splitting iteration methods are applied to solve the problems. Besides, this paper introduces a new kind of matrix splitting, called modulus-based Hermitian and skew-Hermitian matrix relaxation splitting. The corresponding modulus-based matrix splitting iteration method is shown superior to the nonsmooth Newton's method and the existing matrix splitting schemes by some numerical experiments. The convergence analysis is given and validates that our methods are more practical in applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral radius of signless p-Laplacian of a graph, which is a generalization of the quadratic form of the signless Laplacian matrix for p = 2. Let G n , β be the set of simple graphs of order n with a given matching number β. In this paper, the graphs maximizing the largest signless p-Laplacian eigenvalue among G n , β are obtained. [ABSTRACT FROM AUTHOR]

We present a generalization of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated analogue of the SVD which includes features of the Jordan Normal Form, which we show is adequate for our purposes. This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called 'Matrix decompositions over the double numbers'. We adopt the idea presented in that same paper that a pair of matrices is actually a single matrix over the double-complex number system. [ABSTRACT FROM AUTHOR]

Discrete versions of the Laplace and Dirac operators have been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper, we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schrödinger and Dirac equations. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras. [ABSTRACT FROM AUTHOR]

The principal ratio of a graph is the ratio of the greatest and least entry of its principal eigenvector. Since the principal ratio compares the extreme values of the principal eigenvector it is sensitive to outliers. This can be problematic for graphs (networks) drawn from empirical data. To account for this we consider the dispersion of the principal eigenvector (and degree vector). More precisely, we consider the coefficient of variation of the aforementioned vectors, that is, the ratio of the vector's standard deviation and mean. We show how both of these statistics are bounded above by the same function of the principal ratio. Further, this bound is sharp for regular graphs. The goal of this paper is to show that the coefficient of variation of the principal eigenvector (and degree vector) can converge or diverge to the principal ratio in the limit. In doing so, we find an example of a graph family (the complete split graph) whose principal ratio converges to the golden ratio. We conclude with conjectures concerning extremal graphs of the aforementioned statistics. [ABSTRACT FROM AUTHOR]

Let G be a graph, and let $ A(G) $ A (G) be the adjacency matrix of G. The computation of permanent of $ A(G) $ A (G) is #p-complete. Computing permanent of $ A(G) $ A (G) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph $ K_{p, p} $ K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from $ K_{p, p} $ K p , p with six edges deleted are determined by their permanental polynomials. [ABSTRACT FROM AUTHOR]

On the Waring's problem for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as sums of kth powers with $ k\in \{2,3,4,5,6,7,8\} $ k ∈ { 2 , 3 , 4 , 5 , 6 , 7 , 8 } in the literature. In this paper, we provide similar conditions for matrices written as sums of kth powers with $ k\in \{9,10,11,12,13,14,15,16\} $ k ∈ { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 }. [ABSTRACT FROM AUTHOR]

Let v, k, t be positive integers and let $ G(v,k,t) $ G (v , k , t) be the uniform subset graph, defined by all k-subsets of a v-set as its vertices with edges connecting all k-subsets intersecting exactly at t elements. In this paper, we give non-trivial dominating sets for $ G(v,k,2) $ G (v , k , 2) -graphs and their complements for k = 3, 4, 5, 6. The given domination numbers of some of these graphs are sharper than the known bounds. [ABSTRACT FROM AUTHOR]