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Let k and n be two coprime positive integers. In this paper, we show that any $ n\times n $ n × n k-circulant operator matrix $ A_{k,n} $ A k , n with the first row $ A_{1},\dots,A_{n}\in B(H) $ A 1 , ... , A n ∈ B (H) is unitarily equivalent to a generalized permutation operator matrix, an operator matrix that has at most one nonzero entry in each row and in each column. Among other applications in the subject of invertibility, eigenvalues and eigenvectors, using this unitary equivalence in the particular cases of $ k^{2}\pm 1=n $ k 2 ± 1 = n $ ({\rm mod} n) $ (mod n) , we give the following new formulas for computing the numerical radius of these operators: \[ w(A_{k,n})= \begin{cases} \frac{1}{2}\displaystyle \max_{1\leq i\leq n} \sup_{\theta \in \mathbb{R}} \left\| \begin{bmatrix} {\rm e}^{\mathrm{i}\theta}B_{i+1} & B^{*}_{ki+1}\\ B^{*}_{-ki+1} & {\rm e}^{\mathrm{i}\theta}B_{-i+1} \end{bmatrix} \right\|, & {\rm if} \ k^{2}+1\\ & \quad =n \ ({\rm mod}\,n)\\ \displaystyle\max_{1\leq i \leq n}w \left(\begin{bmatrix} 0 & B_{k(i-1)+1}\\ B_{i} & 0 \end{bmatrix} \right), & {\rm if} \ k^{2}-1\\ & \quad =n \ ({\rm mod}\,n), \end{cases} \] w (A k , n) = { 1 2 max 1 ≤ i ≤ n sup θ ∈ R ‖ [ e i θ B i + 1 B ki + 1 ∗ B − ki + 1 ∗ e i θ B − i + 1 ] ‖ , if k 2 + 1 = n (mod n) max 1 ≤ i ≤ n w ([ 0 B k (i − 1) + 1 B i 0 ]) , if k 2 − 1 = n (mod n) , where $ B_{j}=\sum _{s=1}^{n}\omega ^{(s-1)(j-1)}A_{s} $ B j = ∑ s = 1 n ω (s − 1) (j − 1) A s and $ \omega ={\rm e}^{\frac {2\pi \mathrm {i}}{n}} $ ω = e 2 π i n , the nth root of unity. [ABSTRACT FROM AUTHOR]

We present an approach yielding closed analytical formulas for the Kirchhoff indices of circulant graphs with even and odd vertex valency respectively and the prism-like graphs based on circulant graphs. Inspecting the asymptotics of the Kirchhoff index we show that in each of the above-mentioned cases the index can be expressed as the sum of a cubic polynomial and an exponentially vanishing remainder term. [ABSTRACT FROM AUTHOR]

In this paper, we give some new bounds for the eigenvalues of non-monic operator polynomials by applying several numerical radius inequalities to the Frobenius companion matrices of these polynomials. Our bounds refine the existing bounds for monic matrix polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of sets namely analytically imaged sets in the space of m x n matrices. A sufficient condition is obtained for an analytically imaged subset of the set of all n x n matrices to have a dense subset in terms of algebraic multiplicities of the eigenvalues. Also, the counterparts of this result have been studied for singular values of rectangular matrices and it has been shown that all the results hold for convex subsets of matrices. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new estimate for the (product) γ -diagonally dominant degree of the Schur complement of matrices. As applications we discuss the localization of eigenvalues of the Schur complement and present several upper and lower bounds for the determinant of strictly γ -diagonally dominant matrices, which generalizes the corresponding results of Liu and Zhang (SIAM J. Matrix Anal. Appl. 27 (2005) 665-674). [ABSTRACT FROM AUTHOR]

In this paper, we define extensions of the classical eigenvalues of the matrix A ∈ Mm(C). These extensions are eigenvalues matrices for the block matrix A ∈ Mm(Mn), where Mm(Mn) is the set of all m×m block complex matrices with each block in Mn(C), they are.....Among other equalities and inequalities, we prove equalities which relate our new definitions with tr1(A) and tr2(A) as follows,... These relations are extensions of the classical relation tr(A) = Σmh=1λh(A), where A ∈Mm(C). Several new relations and properties of our new definitions are also given. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues. [ABSTRACT FROM AUTHOR]

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the entries that are largest in their row or column, and replacing the remaining entries by zero. This matrix can be interpreted as an opposite to the adjacency matrix, which is on the contrary obtained from the distance matrix by keeping only the entries equal to 1. In the paper, we determine graphs having the second largest eigenvalue of eccentricity matrix less than 1. [ABSTRACT FROM AUTHOR]

Distance or dissimilarity matrices are widely used in applications. We study the relationships between the eigenvalues of the distance matrices and outliers and show that outliers affect the pairwise distances and inflate the eigenvalues. We obtain the eigenvalues of a distance matrix that is affected by k outliers and compare them to the eigenvalues of a distance matrix with a constant structure. We show a discrepancy in the sizes of the eigenvalues of a distance matrix that is contaminated with outliers, present an algorithm and offer a new outlier detection method based on the eigenvalues of the distance matrix. We compare the new distance-based outlier technique with several existing methods under five distributions. The methods are applied to a study of public utility companies and gene expression data. [ABSTRACT FROM AUTHOR]

We present a technique that can be used in the study of a conjecture by Danilov and Koshevoy, concerning triples (A, B,C) of n x n complex selfadjoint matrices such that C = A + B. The conjecture proposes an explicit formula, in terms of traces of compressions of A and B, for one associated hive. We also use this technique to show why an earlier attempt to prove the conjecture fails for n = 4. AMS 2020 Subject Classification: 15A18. [ABSTRACT FROM AUTHOR]

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 $ + 1 (Parter); no change (neutral); and $ -1 $ − 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, both for Hermitian matrices and general matrices. Then, we apply the new ideas, both to classify pendent edges in general graphs, and to understand the existence of 2-downer edge cycles in general graphs, when there is a 2-downer edge. This is a further explanation of why such edges cannot occur in trees. [ABSTRACT FROM AUTHOR]

This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is inf X ⁡ tr (A ˆ X H A X) subject to B ˆ X H B X = I (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is established in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kovač-Striko and Veselić (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix. [ABSTRACT FROM AUTHOR]

The spectral decomposition of a real skew-symmetric matrix is shown to be equivalent to a specific structured singular value decomposition (SVD) of the matrix. Based on such equivalence, we propose a skew-symmetric Lanczos bidiagonalization (SSLBD) method to compute extremal singular values and the corresponding singular vectors of the matrix, from which its extremal conjugate eigenpairs are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of the computed left and right Lanczos basis vectors and the semi-biorthogonality of two sets of basis vectors are needed to compute the singular values accurately and to make the method work as if it does in exact arithmetic. A commonly used efficient partial reorthogonalization strategy is adapted to maintain the desired semi-orthogonality and semi-biorthogonality. For practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

As an extension of Sylvester's matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form. [ABSTRACT FROM AUTHOR]

Frank matrix is one of the popular test matrices for eigenvalue routines because it has well-conditioned and poorly conditioned eigenvalues. In this paper, we investigate the bounds for the maximum eigenvalues of the special cases of the generalized Frank matrices which are called Fibonacci-Frank and Lucas-Frank matrices. Then, we obtain the Euclidean norms and the upper bounds for the spectral norms of these matrices. [ABSTRACT FROM AUTHOR]

In this paper we investigate complex projections A and B so that AB is a diagonalizable matrix. Particularly, we provide necessary and/or sufficient conditions so that AB is a diagonalizable matrix with its eigenvalues belonging to the real segment [0, 1]. Moreover, we investigate on eigenspaces and eigenvalues of the product of two projections. [ABSTRACT FROM AUTHOR]

The eccentricity matrix ℰ (G) = ( u v) of a graph G is constructed from the distance matrix by keeping each row and each column only the largest distances with u v = d (u , v) , if d (u , v) = min { (u) , (v) } , 0 , otherwise , where d (u , v) is the distance between two vertices u and v , and (u) = max { d (u , v) | v ∈ V (G) } is the eccentricity of the vertex u. The ℰ -eigenvalues of G are those of its eccentricity matrix. In this paper, employing the well-known Cauchy Interlacing Theorem we give the following lower bounds for the second, the third and the fourth largest ℰ -eigenvalues by means of the diameter d of G : ξ 2 (G) ≥ − 1 , if d ≤ 2 ; α d , if d ≥ 3 , ξ 3 (G) ≥ − d , and ξ 4 (G) ≥ − 1 − 5 2 d , where α = 0. 3 1 1 1 + is the second largest root of x 3 − x 2 − 3 x + 1 = 0. Moreover, we further discuss the graphs achieving the above lower bounds. [ABSTRACT FROM AUTHOR]

. We consider a minimal realization of a rational matrix. We perturb all the coefficients of matrix polynomial and some coefficients from the realization part present in the realization form of rational matrix. We derive explicit computable formulae for backward error of approximate eigenvalues and eigenpairs of regular rational matrix. We also determine minimal perturbations for all the coefficients of matrix polynomial and some coefficients from the realization part for which approximate eigenvalues are exact eigenvalues of the perturbed rational matrix. [ABSTRACT FROM AUTHOR]

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In this paper we expressed the eigenvalues of a sort of heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From the prescribed eigenvalues, we computed eigenvectors for these types of matrices, giving also a formula not dependent on any unknown parameter for its determinant and inverse. Potential applications of the results are still provided. [ABSTRACT FROM AUTHOR]

In this work, we study the -circulant matrix such that the entries of are or , where and are -Mersenne and -Mersenne–Lucas numbers, respectively. We obtain the eigenvalues and determinants for the matrices and some important identities for the -Mersenne and -Mersenne–Lucas numbers. Furthermore, we find norms and bounds estimation for the spectral norm for these -circulant matrices. [ABSTRACT FROM AUTHOR]

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix. [ABSTRACT FROM AUTHOR]

We study hyper dual numbers and hyper dual number matrices. Firstly, we give some basic properties of dual numbers and hyper dual numbers. Next, we investigate hyper dual number matrices using the properties of dual number matrices. Then we define the dual adjoint matrix of hyper dual matrices and describe some of their properties. [ABSTRACT FROM AUTHOR]

The paper presents methods for the eigenvalue localisation of regular matrix polynomials, in particular, the stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several seconds- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable. [ABSTRACT FROM AUTHOR]

The minimum edge dominating energy of a graph G is defined as the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of G. In this paper, for some finite symmetric groups Γ and subset S of Γ, the minimum edge dominating energy of the Cayley graph of the group Γ, denoted by Cay(Γ, S), is investigated. [ABSTRACT FROM AUTHOR]

In the present paper, we study Min matrix A m i n = [ a m i n (i , j) ] i , j = 1 n , where a s 's are the elements of a real sequence { a s } . We first obtain a recurrence relation for the characteristic polynomial for matrix A m i n , and some relations between the coefficients of its characteristic polynomial. Next, we show that the sequence of the characteristic polynomials of the i × i (i ≤ n) Min matrices satisfies the Sturm sequence properties according to different required conditions of the sequence { a s } . Using Sturm's Theorem, we get some results about the eigenvalues, such as the number of eigenvalues in an interval. Thus, we obtain the number of positive and negative eigenvalues of Min matrix A m i n . Finally, we give an example to illustrate our results. In the present paper, we study Min matrix , where 's are the elements of a real sequence . We first obtain a recurrence relation for the characteristic polynomial for matrix , and some relations between the coefficients of its characteristic polynomial. Next, we show that the sequence of the characteristic polynomials of the Min matrices satisfies the Sturm sequence properties according to different required conditions of the sequence . Using Sturm's Theorem, we get some results about the eigenvalues, such as the number of eigenvalues in an interval. Thus, we obtain the number of positive and negative eigenvalues of Min matrix . Finally, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

The study of constraining the eigenvalues of the sum of two symmetric matrices, say P + Q , in terms of the eigenvalues of P and Q , has a long tradition. It is closely related to estimating a lower bound on the minimum singular value of a matrix, which has been discussed by a great number of authors. To our knowledge, no study has yielded a positive lower bound on the minimum eigenvalue, λ min (P + Q) , when P + Q is symmetric positive definite with P and Q singular positive semi-definite. We derive two new lower bounds on λ min (P + Q) in terms of the minimum positive eigenvalues of P and Q. The bounds take into account geometric information by utilizing the Friedrichs angles between certain subspaces. The basic result is when P and Q are two non-zero singular positive semi-definite matrices such that P + Q is non-singular, then λ min (P + Q) ≥ (1 − cos ⁡ θ F) min ⁡ { λ min (P) , λ min (Q) } , where λ min represents the minimum positive eigenvalue of the matrix, and θ F is the Friedrichs angle between the range spaces of P and Q. Such estimates lead to new lower bounds on the minimum singular value of full rank 1 × 2 , 2 × 1 , and 2 × 2 block matrices. Some examples provided in this paper further highlight the simplicity of applying the results in comparison to some existing lower bounds. [ABSTRACT FROM AUTHOR]

The q-Toda equation is derived from replacing ordinary derivatives with q-derivatives in the famous Toda equation. In this paper, we associate an extension of the q-Toda equation with matrix eigenvalue problems, and then show applications of its time-discretization to computing matrix eigenvalues. With respect to the Lotka–Volterra system, we also have the similar discussion on the case of the Toda equation. [ABSTRACT FROM AUTHOR]

We present new closed-form formulas for the matrix logarithm. Our method is direct and elementary, and it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Using these results, an elegant explicit formula for the principal matrix logarithm can be obtained. Several special cases and examples are stated to illustrate the method presented in this paper. [ABSTRACT FROM AUTHOR]

The anti-adjacency (aka eccentricity) matrix A (G) of a graph G is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. The motivation for this paper is to find answers to two problems: "For which connected graphs the eccentricity (anti-adjacency) matrix is either reducible or irreducible?" and "Characterize the graphs with a small number of distinct anti-adjacency eigenvalues". In this context, we get the sufficient and necessary conditions of the graphs which have an irreducible matrix. For the second problem, we characterize the graphs with exactly two distinct eigenvalues different from 0. We also obtain some spectral properties of the anti-adjacency matrix. [ABSTRACT FROM AUTHOR]

For a simple finite graph G, the generalized adjacency matrix is defined as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ] , where A (G) and D (G) are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The A α -spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the A α (G) . In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the A α -spread of a graph. Electron. J. Linear Algebra2020, 36, 214–227). Furthermore, we show that the path graph, P n , has the smallest S (A α) among all trees of order n. We establish a relationship between S (A α) and S (A). We obtain several bounds for S (A α) . [ABSTRACT FROM AUTHOR]

We classify maximal digraphs whose Hermitian spectral radius is at most 2. [ABSTRACT FROM AUTHOR]

We show some upper bounds for the product of arbitrarily selected singular values of the sum of two matrices. The results are additional to our previous work on the lower bound eigenvalue inequalities of the sum of two positive semidefinite matrices. Besides, we state explicitly Hoffman's minimax theorem with a proof, and as applications of our main results, we revisit and give estimates for related determinant inequalities of Hua type. [ABSTRACT FROM AUTHOR]

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matrices. [ABSTRACT FROM AUTHOR]

The notion of root polynomials of a polynomial matrix P (λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R (λ) , possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R (λ) , without any need to assume that R (λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R (λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ 0. If λ 0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations. [ABSTRACT FROM AUTHOR]

Many problems in applied mathematics are solved by computing the eigenvalues, singular values, spectral and Euclidean norms of the skew-symmetric matrices. In this article, we first introduce the eigenvalues and singular values of the matrix A = [xi - xj ]in,j=1. Then we obtain the spectral norm and Euclidean norms of A. Finally, some numerical examples are taken to illustrate the correctness of the concluded results. [ABSTRACT FROM AUTHOR]

In this paper, G-majorization inequalities are proven for the gradients of two Gateaux differentiable G-increasing functions defined on a real linear space endowed with the structure of an Eaton triple. Such inequalities can be interpreted as a monotonicity property of certain operators and functionals with respect to a special preorder on the class of all G-increasing functions. The notion of a c-strongly G-increasing function is introduced and used to illustrate the derived inequalities. The obtained results are also applied for matrix Eaton triples connected with the eigenvalues map on the space of Hermitian matrices as well as with the singular values map on the space of complex matrices. [ABSTRACT FROM AUTHOR]

A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs. [ABSTRACT FROM AUTHOR]

In this paper, we present the γ (product γ )-diagonally dominant degree on the Schur complement of γ (product γ )-diagonally dominant matrices. Consequently, the disc theorems for the Schur complement of γ (product γ )-diagonally dominant matrices are obtained using the diagonally dominant degree on Schur complements, which improves and extends some related results. Further, to solve large-scale linear systems, we give the spectral radius estimates for the inverse matrices of γ (product γ )-diagonally dominant matrices and their Schur complements. When the coefficient matrix of linear system is γ (product γ )-diagonally dominant matrix, according to the special properties and the structures of γ (product γ )-diagonally dominant matrices, we design a class of special iteration for solving equations. Combining the spectral iteration with the Schur-based iterative method, we obtain a new iterative method called the Schur-based inverse algorithm, which reduces the order of the matrix and has a good convergence. The numerical examples are given to verify the validity of our results. [ABSTRACT FROM AUTHOR]

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n × n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2 × 2 and the irreducible 3 × 3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) = { v 1 , ... , v n } and E P (G) be an n × n matrix whose (i , j) -entry is the maximum number of internally edge-disjoint paths between v i and v j , if i ≠ j , and zero otherwise. Also, define E P ¯ (G) = E P (G) + D , where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing v i , whose E P (G) is a multiple of J−I. Among other results, we determine the spectrum and the energy of the matrix E P ¯ (G) for an arbitrary bicyclic graph G. [ABSTRACT FROM AUTHOR]

The article discusses extensions of the results related to bounds for the eigenvalues and spreads of matrices using traces.

G is a simple connected graph with adjacency matrix A(G) and degree diagonal matrix D(G). The signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2017, Nikiforov [1] defined the matrix Aα(G) = α D(G) + (1 − α)A(G) for α ∈ [0,1]. The Aα-spectral radius of G is the maximum eigenvalue of Aα (G). In 2019, Liu et al. [2] defined the matrix Θk(G) as Θk (G) = kD(G) + A(G), for k ∈ ℝ. In this paper, we present a new type of lower bound for the Aα-spectral radius of a graph after vertex deletion. Furthermore, we deduce some corollaries on Θk (G), A(G), Q(G) matrices. [ABSTRACT FROM AUTHOR]

• The mathematical model in this paper includes both DC/DC and AC/DC converters. • The optimization parameters of matrix perturbation theory comprehensively consider the small perturbation stability, damping ratios and stability margins objective function. • Matrix perturbation theory can avoid the repeated formation of the system state matrix and the repeated calculation of the characteristic solution problem. • Droop coefficient optimization is an effective method to improve the stability of the system with small perturbations. This paper proposes a perturbation analysis and optimization method for the droop coefficient of multi-terminal DC distribution systems to address their weak damping and instability. Firstly, a small disturbance model of the multi-terminal DC distribution system with droop control is established. A proposed objective function for optimizing droop coefficients in DC/DC and AC/DC converters comprehensively considers factors such as small disturbance stability, damping ratio, and stability margin. Then, the eigenvalues obtained from the droop coefficients perturbation analysis of each sub-matrix module of the system are used in the optimization iteration to prevent the system matrix from being repeatedly formed and the eigenvalues from being repeatedly solved during the optimization process. Finally, the model is built in PLECS with RT-Box for experimental verification. The results demonstrate that this method can effectively enhance the stability of DC distribution systems. [ABSTRACT FROM AUTHOR]

An Infinite and finite dimensional generalized Hilbert tensor with a is positive definite if and only if a > 0. The infinite dimensional generalized Hilbert tensor related operators F ∞ and T ∞ are bounded, continuous and positively homogeneous. A generalized Cauchy tensor of which generating vectors are c , d is positive definite if and only if every element of vector d is not zero and each element of vector c is positive and mutually distinct. The 4th order n -dimensional generalized Cauchy tensor is matrix positive semi-definite if and only if every element of generating vector c is positive. Finally, the other properties of generalized Cauchy tensor are presented. [ABSTRACT FROM AUTHOR]

Let A,B,C ∈ Cn×n be positive semidefinite matrices and let |A|, |B|, |C| be determinants of A, B, C ∈ Cn×n respectively. In this paper, the authors prove two determinantal inequalities |A + B + C| + |C| ≥ |A + C| + |B + C| + (2n - 2) |AB|1/2 + 3(3n-1 - 2n + 1) |ABC|1/3 and |A + B + C| + |A| + |B| + |C| ≥ |A + B| + |A + C| + |B + C| + 3(3n-1 - 2n + 1)|ABC|1/3. These two inequalities refine known ones. [ABSTRACT FROM AUTHOR]

Waiving the positive definiteness of the leading matrix A in a hyperbolic quadratic eigenvalue problem Q(λ)x=(λ²A+λB+C)x=0, x 6ǂ0 one obtains a definite eigenvalue problem, which is known to have 2n eigenvalues in R?{⃵}. One of the characterizations of the definite quadratic eigenvalue problem is the existence of parameters ξ and µ so that Q(µ) is positive definite and Q(x) is negative definite, where ξ and µ are not known in advance. In this paper we consider the impact of the properties of the stiffness matrix C of the quadratic pencil Q(λ) on the corresponding definite quadratic eigenvalue problem and on the localization of the parameters ξ and µ. [ABSTRACT FROM AUTHOR]

We prove a spectral radius inequality for operator matrices with commuting entries. Applications of this inequality to the polynomial eigenvalue problem are also given. [ABSTRACT FROM AUTHOR]

We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we deal with some classical eigenvalue inequalities for Hermitian matrices, including the Cauchy interlacing theorem and the Weyl inequality, in a simple and unified approach. We also give a common generalization of eigenvalue inequalities for (Hermitian) normalized Laplacian matrices of simple (signed, weighted, directed) graphs. Our approach is also suitable for Hermitian matrices of the second kind of digraphs recently introduced by Mohar. [ABSTRACT FROM AUTHOR]