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In [1] , the set of refined inertias H n for sign patterns is expanded to H n ⁎ for zero-nonzero patterns. Also, it is proved that there are no zero-nonzero patterns of even order 2 m that require H 2 m ⁎ . In this paper, it is shown that there are no zero-nonzero patterns of odd order n ≥ 5 that require H n ⁎ , which together with the previous result prove that there are no zero-nonzero patterns of order n ≥ 4 that require H n ⁎ . [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The distance Laplacian matrix of G is defined as D L ( G ) = T r ( G ) − D ( G ) , where D ( G ) is the distance matrix and T r ( G ) = diag ( t r v 1 , t r v 2 , … , t r v n ) is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of D L ( G ) is called the distance Laplacian spectral radius of G . In this paper, we obtain a graft transformation of a connected graph, which increases its distance Laplacian spectral radius. Using this transformation, we prove a conjecture involving the distance Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

For any algebraically closed field F and any two square matrices A , B over F , Sylvester (1884) [8] and Cecioni (1910) [1] showed that A X = X B implies X = 0 if and only if A and B have no common eigenvalue. It is proved that a third equivalent statement is that, for any given polynomials f , g in F [ t ] , there exists h in F [ t ] such that f ( A ) = h ( A ) and g ( B ) = h ( B ) . Corresponding results hold also for any finite set of square matrices over F , and these lead to a new property of all associative rings and algebras (even over arbitrary fields) with 1. [ABSTRACT FROM AUTHOR]

Let K n be the set of all n × n lower triangular ( 0 , 1 ) -matrices with each diagonal element equal to 1, L n = { Y Y T : Y ∈ K n } and let c n be the minimum of the smallest eigenvalue of Y Y T as Y goes through K n . The Ilmonen–Haukkanen–Merikoski conjecture (the IHM conjecture) states that c n is equal to the smallest eigenvalue of Y 0 Y 0 T , where Y 0 ∈ K n with ( Y 0 ) i j = 1 − ( − 1 ) i + j 2 for i > j . In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices. [ABSTRACT FROM AUTHOR]

In this paper we prove a new characterization of the max-plus singular values of a max-plus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singular values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially parameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter independent classical matrices. We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the 2-norm condition number and that this action can be explained using our new theory for max-plus singular values. [ABSTRACT FROM AUTHOR]

We give new proofs of several fundamental results of spectral theory in max algebra. This includes the description of the spectrum in max algebra of a given non-negative matrix via local spectral radii, the spectral theorem and the spectral mapping theorem in max algebra. The latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra. Our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non-negative matrices. [ABSTRACT FROM AUTHOR]

A new tensor eigenvalue inclusion set is given, and proved to be tighter than those in L.Q. Qi (2005) [18] and C.Q. Li, Y.T. Li, X. Kong (2014) [12] . In addition, we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by Qi (2005) for weakly irreducible tensors. As applications, we give new bounds for the spectral radius of nonnegative tensors, some sufficient conditions for a tensor to be a strong M -tensor and some inequalities to identify the positive definiteness for an even-order real supersymmetric tensor. [ABSTRACT FROM AUTHOR]

The distance matrix of a simple graph G is D ( G ) = ( d i j ) , where d i j is the distance between i th and j th vertices of G . The spectrum of the distance matrix is known as the distance spectrum or D -spectrum of G . A simple connected graph G is called distance regular if it is regular, and if for any two vertices x , y ∈ G at distance i , there are constant number of neighbors c i and b i of y at distance i − 1 and i + 1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d + 1 distinct D -eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D -eigenvalues. We also prove that distance regular graphs satisfying b i = c d − 1 have at most ⌈ d 2 ⌉ + 2 distinct D -eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16] . [ABSTRACT FROM AUTHOR]

From the parameters ( n , k , t , λ , μ ) of a directed strongly regular graph (dsrg) A. Duval (1988) [4] showed how to compute the eigenvalues and multiplicities of the adjacency matrix, and thus the rank of the adjacency matrix. For every rational number q , where 1 5 ≤ q ≤ 7 10 , there is a feasible (i.e., satisfying Duval's conditions) parameter set for a dsrg with rank 5 and with k n = q . In this paper we show that there exist a dsrg with such a feasible parameter set only if k n is 1 5 , 1 3 , 2 5 , 1 2 , 3 5 , or 2 3 . Every dsrg with rank 5 therefore has parameters of a known graph. The proof is based on an enumeration of 5 × 5 matrices with entries in { 0 , 1 } . [ABSTRACT FROM AUTHOR]

A P-set of a symmetric matrix A is a set α of indices such that the nullity of the matrix obtained from A by removing rows and columns indexed by α is | α | more than the nullity of A . It is known that each subset of a P-set is a P-set. It is also known that a set of indices such that each singleton subset is a P-set need not be a P-set. This note shows that if all pairs of vertices of a set with at least two elements are P-sets, then the set is a P-set. [ABSTRACT FROM AUTHOR]

In this study we investigate the monotonic behavior of the smallest eigenvalue t n and the largest eigenvalue T n of the n × n matrix E n T E n , where the ij -entry of E n is 1 if j | i and 0 otherwise. We present a proof of the Mattila–Haukkanen conjecture which states that for every n ∈ Z + , t n + 1 ≤ t n and T n ≤ T n + 1 . Also, we prove that lim n → ∞ ⁡ t n = 0 and lim n → ∞ ⁡ T n = ∞ and we give a lower bound for t n . [ABSTRACT FROM AUTHOR]

Connected graphs with the same number of vertices and edges are called unicyclic graphs. All unicyclic graphs whose second largest eigenvalue does not exceed 1, denoted by U ( λ 2 ) , are determined by Xu [10] . Let G t , s denote the graph obtained by attaching t ≥ 1 pendent paths of length 2 and s ≥ 0 pendent edges to a vertex of a triangle. In this paper, we first prove that the graph G t , s is determined by its adjacency spectrum. Then we prove that the graphs in U ( λ 2 ) are determined by their adjacency spectra except for four graphs. [ABSTRACT FROM AUTHOR]

This paper presents applications of a remarkable majorization inequality due to Bapat and Sunder in three different areas. The first application is a proof of Hiroshima's 2003 result which arises in quantum information theory. The second one is an extension of some eigenvalue inequalities that have been used to bound chromatic number of graphs. The third application is a simplified proof of a majorization inequality from the analysis of distributed Kalman filtering. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G , defined as Q ( G ) = Tr ( G ) + D ( G ) , where Tr ( G ) is the diagonal matrix of vertex transmissions of G and D ( G ) is the distance matrix of G . In [10] , Xing et al. determined the graph with minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. For n ≥ 3 , let T n − 3 , 1 1 be the n -vertex tree of maximum degree n − 2 . In this paper, we show that T n − 3 , 1 1 gives the second minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. Moreover, we prove two conjectures involving the second largest eigenvalue of the distance signless Laplacian matrix Q ( G ) of graph G . [ABSTRACT FROM AUTHOR]

In this paper, we prove various eigenvalue and trace inequalities of objects in the setting of simple Euclidean Jordan algebras via majorization techniques. [ABSTRACT FROM AUTHOR]

Abstract: Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion–Hoffman theorem is given. [Copyright &y& Elsevier]

Abstract: We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both the standard combinatorial Laplacian and the renormalized Laplacian. We also provide upper bounds for sums of squares of eigenvalues of these three matrices. Among our results, we generalize an inequality of Fiedler for the extreme eigenvalues of the graph Laplacian to a bound on the sums of the smallest (or largest) k such eigenvalues, . Furthermore, if are the eigenvalues of the graph Laplacian , in increasing order, on a finite graph with vertices and edges which is isomorphic to a subgraph of the ν-dimensional infinite cubic lattice, then the spectral sums obey a Weyl-type upper bound, a simplification of which reads for each . This and related estimates for provide a family of necessary conditions for the embeddability of the graph in a lattice of dimension ν or less. [Copyright &y& Elsevier]

Abstract: We prove that heat-bath chains (which we define in a general setting) have no negative eigenvalues. Two applications of this result are presented: one to single-site heat-bath chains for spin systems and one to a heat-bath Markov chain for sampling contingency tables. Some implications of our main result for the analysis of the mixing time of heat-bath Markov chains are discussed. We also prove an alternative characterisation of heat-bath chains, and consider possible generalisations. [Copyright &y& Elsevier]

Abstract: Several matrix norms of the classical Frobenius companion matrices of a monic polynomial have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots λ of . Recently, M. Fiedler (2003) [9] has introduced a new family of companion matrices of that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on . The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1- and ∞-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials. [Copyright &y& Elsevier]

Abstract: For a distance matrix , its inertia is the triple of integers , where , , denote the number of positive, 0, negative eigenvalues of , respectively. The D-energy is the sum of the absolute eigenvalues of . In this paper, we first study the inertia of distance matrices of complete k-partite graphs; Then, as applications, we not only prove a conjecture proposed by Caporossi et al. (2009) in [5] in a different way from Stevanović et al. (2013) [11] but also obtain the formula of the D-energy of the remaining complete k-partite graphs. At last, we obtain the graphs with the maximum (resp. minimum) D-energy among all the complete k-partite graphs with n vertices. [Copyright &y& Elsevier]

Abstract: In 2005, Lu, Liu and Tian [2] published two inequalities involving the domination number and Laplacian eigenvalues of graphs. A short proof of their results is given, an error is pointed out, and improved inequalities are presented. [Copyright &y& Elsevier]

Abstract: Let G be a simple connected graph on n vertices and m edges. Lima et al. (2011) in [2] posed the following conjecture on the least eigenvalue of the signless Laplacian of G: . In this paper we prove a stronger result: For any graph with n vertices and m edges, we have . [Copyright &y& Elsevier]

Abstract: Let be a connected signed graph, where G is the underlying simple graph and is the sign function on the edges of G. Let , be the Laplacian of Γ and be its eigenvalues. It is well-known that a signed graph Γ is balanced if and only if . Here we show that, if Γ is unbalanced, then estimates how much Γ is far from being balanced. In particular, let (resp. ) be the frustration number (resp. frustration index), that is the minimum number of vertices (resp. edges) to be deleted such that the signed graph is balanced. Then we prove that Further we analyze the case when . In the latter setting, we identify the structure of the underlying graph G and we give an algebraic condition for which leads to the above equality. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph with n vertices and edges, and be the signless Laplacian eigenvalues of G. Let , where . F. Ashraf et al. conjectured that for . In this paper, we give various upper bounds for , and prove that this conjecture is true for the following cases: connected graph with sufficiently large k, unicyclic graphs and bicyclic graphs for all k, and tricyclic graphs when . Finally, we discuss whether the upper bound given in this conjecture is tight or not for c-cyclic graphs and propose some problems for future research. [Copyright &y& Elsevier]

Abstract: In this article we study the index (the largest eigenvalue of the adjacency matrix) in two special classes of trees, namely: starlike trees and double brooms. For each class, we determine conditions for the index to be integer. We examine conditions in order to compare the index of a starlike tree with , where Δ is the maximum degree of the graph. We also prove that indices of a broom-like tree and a double broom, except for some cases, are between and , characterizing when occurs one equality. Furthermore, we build in these classes, infinite families of non-integral trees with integer index. [Copyright &y& Elsevier]

Abstract: Let G be a finite graph with H as a star complement for an eigenvalue other than 0 or −1. Let , denote respectively the vertex-connectivity and minimum degree of G. We prove that is controlled by and . In particular, for each there exists a smallest non-negative integer such that whenever and . We show that , , , and . [Copyright &y& Elsevier]

Abstract: Let be a nonnegative matrix with . We prove some lower bounds for the spread of A that is defined as the maximum distance between any two eigenvalues of A. If A has only two distinct eigenvalues, then , where is the spectral radius of A. Moreover, this lower bound is the best possible. [Copyright &y& Elsevier]

Abstract: We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any of its principal submatrices) for the class of matrices introduced by Kotelyansky (all principal and almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices possess this property. We also prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices. [Copyright &y& Elsevier]

Abstract: In this paper we present an upper bound on the degree of a regular graph with girth at least 5 in terms of its second largest eigenvalue (usually denoted by ). Next, we consider bipartite r-regular graphs with girth at least 6 whose second largest eigenvalue satisfies and prove that such graphs are the incidence graphs of two-class partially balanced incomplete block designs and also balanced incomplete block designs. Upper bound on the order of such graphs is also given. We also consider the relations between the incidence graphs of two-class partially balanced incomplete block designs and distance-regular graphs. In particular, we determine all regular reflexive graphs of degree 3 with girth at least 5, all regular graphs with girth at least 5 satisfying and all bipartite regular reflexive graphs with girth at least 6. [Copyright &y& Elsevier]

Abstract: We investigate the application of the LR Cholesky algorithm to symmetric hierarchical matrices, symmetric simple structured hierarchical matrices and symmetric hierarchically semiseparable (HSS) matrices. The data-sparsity of these matrices make the otherwise expensive LR Cholesky algorithm applicable, as long as the data-sparsity is preserved. We will see in an example that the ranks of the low rank blocks grow and the data-sparsity gets lost. We will explain this behavior by applying a theorem on the structure preservation of diagonal plus semiseparable matrices under LR Cholesky transformations. Therefore we have to give a new more constructive proof for the theorem. We will show that the structure of -matrices is almost preserved and so the LR Cholesky algorithm is of almost quadratic complexity for -matrices. [Copyright &y& Elsevier]

Abstract: We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed. [Copyright &y& Elsevier]

Abstract: This article discusses Ritz and harmonic Ritz values computed from a Krylov subspace, generated by a normal matrix. We give necessary and sufficient conditions for a given tuple of complex numbers to represent Ritz values: the existence of a positive solution to a certain linear system with a Cauchy matrix. Using this characterization, we prove several localization results for the Ritz values of normal matrices and discuss consequences for the convergence of the restarted Arnoldi algorithm. Similar results are shown for the harmonic case as well. [Copyright &y& Elsevier]

Abstract: Positive semidefinite matrices partitioned into a small number of Hermitian blocks have a remarkable property. Such a matrix may be written in a simple way from the sum of its diagonal blocks: the full matrix is a kind of average of copies of the sum of the diagonal blocks. This entails several eigenvalue inequalities. The proofs use a decomposition lemma for positive matrices, isometries with complex entries, and the Pauli matrices. [Copyright &y& Elsevier]

Abstract: An explicit solution to the partial eigenvalue assignment problem of high order control systems with time delay is presented. The problem is solved by a direct method without turning high order systems into the first order form. The solution can be implemented with only a partial knowledge of the eigenvalues and the corresponding eigenvectors of the matrix polynomial. Necessary and sufficient conditions for our results are proven. Numerical examples are given to illustrate the effect of the approach. [Copyright &y& Elsevier]

Abstract: ubic graphs of given order with maximum number of triangles are characterized. Consequently, it is proved that certain cubic graphs with maximum number of triangles are determined by their adjacency spectrum. [Copyright &y& Elsevier]

Abstract: We prove several eigenvalue inequalities for the differences of various means of two positive invertible operators A and B on a separable Hilbert space, under the assumption that is compact. Equality conditions of these inequalities are also obtained. [Copyright &y& Elsevier]

Abstract: This paper investigates the relation between the eigenvalues of a Euclidean distance matrix (EDM) and those of the corresponding positive semidefinite matrix. More precisely, let and be two EDMs and let and be the corresponding positive semidefinite matrices such that . In this paper, when the eigenvalues of majorizes those of , a condition under which the eigenvalues of majorizes those of is derived. [Copyright &y& Elsevier]

Abstract: We study the energy (i.e., the sum of the absolute values of all eigenvalues) of so-called tadpole graphs, which are obtained by joining a vertex of a cycle to one of the ends of a path. By means of the Coulson integral formula and careful estimation of the resulting integrals, we prove two conjectures on the largest and second-largest energy of a unicyclic graph due to Caporossi, Cvetković, Gutman and Hansen and Gutman, Furtula and Hua, respectively. Moreover, we characterise the non-bipartite unicyclic graphs whose energy is largest. [Copyright &y& Elsevier]

Abstract: The energy of a simple graph , denoted by , is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let denote the cycle of order and the graph obtained from joining two cycles by a path with its two leaves. Let denote the class of all bipartite bicyclic graphs but not the graph , which is obtained from joining two cycles and ( and ) by an edge. In [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41(2001) 1002–1005], Gutman and Vidović conjectured that the bicyclic graph with maximal energy is , for and . In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427(2007) 87–98], Li and Zhang showed that the conjecture is true for graphs in the class . However, they could not determine which of the two graphs and has the maximal value of energy. In [B. Furtula, S. Radenković, I. Gutman, Bicyclic molecular graphs with the greatest energy, J. Serb. Chem. Soc. 73(4)(2008) 431–433], numerical computations up to were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of is larger than that of , which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open. [Copyright &y& Elsevier]

Abstract: Let G be a simple connected graph with n vertices and m edges. Denote the degree of vertex by . The matrix is called the signless Laplacian of G, where and denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let be the largest eigenvalue of . In this paper, we first present two sharp upper bounds for involving the maximum degree and the minimum degree of the vertices of G and give a new proving method on another sharp upper bound for . Then we present three sharp lower bounds for involving the maximum degree and the minimum degree of the vertices of G. Moreover, we determine all extremal graphs which attain these sharp bounds. [Copyright &y& Elsevier]

Abstract: We highlight some properties of the field of values (or numerical range) of an oblique projector P on a Hilbert space, i.e., of an operator satisfying . If P is neither null nor the identity, we present a direct proof showing that , i.e., the field of values of an oblique projection coincides with that of its complementary projection. We also show that is an elliptical disk (i.e., the set of points circumscribed by an ellipse) with foci at 0 and 1 and eccentricity . These two results combined provide a new proof of the identity . We discuss the influence of the minimal canonical angle between the range and the null space of P, on the shape of . In the finite dimensional case, we show a relation between the eigenvalues of matrices related to these complementary projections and present a second proof to the fact that is an elliptical disk. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertices and be the largest eigenvalue of the adjacency matrix of . We study how large can be when does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems. [Copyright &y& Elsevier]

Abstract: We survey results concerning star complements in finite regular graphs, and note the connection with designs and strongly regular graphs in certain cases. We include improved proofs along with new results on stars and windmills as star complements. [Copyright &y& Elsevier]

Abstract: Let be a natural number and let be a graph with at least vertices. Brouwer conjectured that the sum of the largest Laplacian eigenvalues of is at most , where is the number of edges of . We prove this conjecture for . We also show that if is a tree, then the sum of the largest Laplacian eigenvalues of is at most . [Copyright &y& Elsevier]

Abstract: The energy of a graph is equal to the sum of the absolute values of the eigenvalues of , which in turn is equal to the sum of the singular values of the adjacency matrix of . Let X, Y, and Z be matrices, such that . The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities. [Copyright &y& Elsevier]

Abstract: In this paper, we show that if the second largest eigenvalue of a -regular graph is less than , then the graph is -edge-connected. When is or , we prove stronger results. Let denote the largest root of . We show that if the second largest eigenvalue of a -regular graph is less than , then is -edge-connected and we prove that if the second largest eigenvalue of is less than , then is -edge-connected. [Copyright &y& Elsevier]

Abstract: Real stable matrices and with rank of equal to one have a common Lyapunov solution if and only if their product has no real negative eigenvalue. This was proved by Shorten and Narendra [R.N. Shorten, K.S. Narendra, On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Trans. Automat. Control 48 (4) (2003) 618–621], whose proof is based on the fundamental results of Kalman on Luré’s problem. In this paper we give an alternative proof of this result and its generalization to the general regular inertia case, and to the case when the matrices and are complex. [Copyright &y& Elsevier]

Abstract: In a number of his recent papers Karl Gustafson has outlined the similarities between the Antieigenvalue Theory he founded and several finite dimensional matrix optimization theorems for positive matrices arising in statistics. In this paper, we will show how the techniques that the author and Karl Gustafson have used for computation of Antieigenvalues can also be applied to prove and generalize these matrix optimization theorems in statistics. We will primarily focus on two techniques which we have used in Antieigenvalue computations in recent years. These two techniques are a two nonzero component property for certain class of functionals, and converting the matrix optimization problems in statistics to a convex programing problem. Indeed, these two techniques allow us to generalize some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces. [Copyright &y& Elsevier]

Abstract: Let be a differential field whose field of constants is algebraically closed and let A be a matrix with coefficients in which commutes with its derivative DA. We show that all the eigenvalues of A lie in , answering open problem 22 of [1]. We also give a simple proof of a theorem of Schur characterizing matrices A with the property that the derivatives of A of all orders mutually commute. [Copyright &y& Elsevier]