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In this paper, we give a characterization of regular two-graphs of order n in terms of spectral data of two-graphs of order n − 1 and n − 2. Moreover, we prove that a two-graph of order n is regular if and only if all its induced sub-two-graphs with n − 2 vertices have the same characteristic polynomial. [ABSTRACT FROM AUTHOR]

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP. [ABSTRACT FROM AUTHOR]

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊ n 2 ⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions { c n gri (A) } n ≥ 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T 2 ⁎ -equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums. [ABSTRACT FROM AUTHOR]

The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in number theory and dynamics. [ABSTRACT FROM AUTHOR]

Let H be a field with Q ⊆ H ⊆ C , and let p (λ) be a polynomial in H [ λ ] , and let A ∈ H n × n be nonderogatory. In this paper we consider the problem of finding a solution X ∈ H n × n to p (X) = A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A = p (X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form. [ABSTRACT FROM AUTHOR]

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Consider a polynomial f (z) = a n z n +... + a 1 z + a 0 of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the p th Hadamard power of f : f [ p ] (z) : = a n p z n +... + a 1 p z + a 0 p is stable if p is a positive integer number. However, it turns out that f [ p ] does not need to be stable for all real p > 1. A counterexample is known for n = 8 and p = 1.139. On the other hand, f [ p ] is stable for n = 1 , 2 , 3 , 4 , and every p > 1. In this paper we fill the gap by showing that f [ p ] is stable for n = 5 and constructing counterexamples for n ≥ 6. Moreover, by means of Rouché's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results. [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered. [ABSTRACT FROM AUTHOR]

In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If L n is an n -dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for Id (L n) , the polynomial identities of L n , and we explicitly compute the images of multihomogeneous polynomials on L n. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial f to be a subspace of L n. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for L n. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case. [ABSTRACT FROM AUTHOR]

Let G be a group and F an infinite field. Assume that A is a finite dimensional F -algebra with an elementary G -grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F -algebra A ⊗ C , where C is an H -graded colour β -commutative algebra. More precisely, under a technical condition, we provide a basis for the T G -ideal of graded polynomial identities of A ⊗ C , up to graded monomial identities. Furthermore, the F -algebra of upper block-triangular matrices U T (d 1 , ... , d n) , as well as the matrix algebra M n (F) , with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤ 2 d − 1 , for M d (E) and M q (F) ⊗ U T (d 1 , ... , d n) , with a tensor product grading, is exhibited. In this latter case, d = d 1 + ... + d n. Here E denotes the infinite dimensional Grassmann algebra with its natural Z 2 -grading, and the grading on M q (F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero. [ABSTRACT FROM AUTHOR]

Given a possibly singular matrix polynomial P (z) , we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space DL (P) introduced in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl. 28(4), 971-1004, 2006] are related to those of P (z). If P (z) is regular, it is known that those pencils in DL (P) satisfying the generic assumptions in the so-called eigenvalue exclusion theorem are strong linearizations for P (z). This property and the block-symmetric structure of the pencils in DL (P) have made these linearizations among the most influential for the theoretical and numerical treatment of structured regular matrix polynomials. However, it is also known that, if P (z) is singular, then none of the pencils in DL (P) is a linearization for P (z). In this paper, we prove that despite this fact a generalization of the eigenvalue exclusion theorem holds for any singular matrix polynomial P (z) and that such a generalization allows us to recover all the relevant quantities of P (z) from any pencil in DL (P) satisfying the eigenvalue exclusion hypothesis. Our proof of this general theorem relies heavily on the representation of the pencils in DL (P) via Bézoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181-209, 2015]. [ABSTRACT FROM AUTHOR]

A graph G is divisible by a graph H if the characteristic polynomial of G is divisible by that of H. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape graph P 2 , 2 ; n − 4 2 , n − 7 , known to be (for n large enough) the minimizer of the spectral radius among the graphs of order n and diameter n − 5 , is determined by its adjacency spectrum if and only if n ≠ 10 , 13 , 15. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L ⁡ (A) coincides with exp ⁡ (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [ABSTRACT FROM AUTHOR]

In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we present a general method to construct q -analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method. [ABSTRACT FROM AUTHOR]

A (k , g) -cage is a k -regular simple graph of girth g with minimum possible number of vertices. In this paper, (k , g) -cages which are Moore graphs are referred as minimal (k , g) -cages. A simple connected graph is called distance regular (DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular (DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal (k , g) -cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal (k , g) -cages we give a formula for distance spectral radius in terms of k and g , and also determine polynomials of degree ⌊ g 2 ⌋ , which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal (k , g) -cage of diameter d has d + 1 distinct distance eigenvalues, and this partially answers a problem posed in [1]. We prove that every DBR graph is a 2-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal (k , g) -cages. Finally we determine the full distance spectrum of subdivision of some minimal (k , g) -cages. [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]

In this paper we study the polynomiality and co-polynomiality of association schemes. These two notions were introduced by Ito. We show that P -polynomial association schemes are co-polynomial and Q -polynomial association schemes are polynomial. We also determine the polynomiality and co-polynomiality for 3-class association schemes. Further we give examples of association schemes that are polynomial but not co-polynomial, or co-polynomial but not polynomial. [ABSTRACT FROM AUTHOR]

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. The n th descending diagonal of the ordinary Riordan matrix and the n th descending diagonal of the exponential Riordan matrix have the generating functions respectively g n (φ x) / (1 − φ x) n + 1 and h n (φ x) / (1 − φ x) 2 n + 1 , where g n (x) , h n (x) are polynomials of degree ≤ n. We will call these polynomials the numerator polynomials of Riordan matrices. General properties of these polynomials were considered in separate paper. In this paper, we will consider numerator polynomials of the Riordan matrices associated with the family of series a (β) (x) = a (x (β) a β (x)). The matrices of transformations, in which these polynomials participate, have the form A n E n β A n − 1 , where A n is the certain matrix of order n + 1 , E is the matrix of the shift operator. The main focus is on studying the properties of these matrices. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) and i , j ∈ V , denote by d i , d j the degrees of vertices i , j in G. Let f (d i , d j) > 0 be a function symmetric in i and j. Define a matrix A f (G) , called the weighted adjacency matrix of G , with the ij -entry A f (G) (i , j) = f (d i , d j) if i ∼ j and A f (G) (i , j) = 0 otherwise. In this paper, we find the extremal trees with the largest radius of A f when f (x , y) is increasing and convex in variable x. We also find the extremal tree with the smallest radius of A f when f (x , y) has a form P (x , y) or P (x , y) , where P (x , y) is a symmetric polynomial with nonnegative coefficients and zero constant term. This paper tries to unify the spectral study of weighted adjacency matrices of graphs weighted by some topological indices. [ABSTRACT FROM AUTHOR]

Let A be a unital C ⁎ algebra and for every a ∈ A , r (a) denote the numerical radius of a ∈ A. The power inequality for numerical radius states that for every polynomial P (z) = z n and a ∈ A the inequality P (r (a)) ≥ r (P (a)) holds. In this paper, we get a characterization of polynomials with real coefficients that satisfy the power inequality on all 2 × 2 matrices with real entries. We also characterize all polynomials that satisfy the power inequality on every commutative unital C ⁎ algebra. [ABSTRACT FROM AUTHOR]

In [19] , Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial P (λ) over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When P (λ) is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to P (λ) , in which the diagonal blocks are of size at most 2 × 2. This paper generalizes these results to regular matrix polynomials P (λ) over arbitrary fields F , showing that any such P (λ) can be quasi-triangularized to a spectrally equivalent matrix polynomial over F of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the F -irreducible factors in the Smith form for P (λ). [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient model reduction method for discrete time-delay systems based on the expansions of systems under Charlier polynomials. Making full use of the properties of Charlier polynomials and the structure of discrete time-delay systems, the projection space built by state variables is embedded in a high-order Krylov subspace. Further, a high-order Krylov subspace method is developed to generate discrete time-delay reduced systems. The proposed method is independent of the choice of inputs since the high-order Krylov subspace sequence does not involve the expansion coefficients of inputs. Besides, theoretical analysis shows that the resulting discrete time-delay reduced system characterizes the property of invariable coefficients. Finally, two numerical examples demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7,10,24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form a x + b 0 x q s + b 1 x q s + n + b 2 x q s + 2 n + ... + b t − 1 x q s + n (t − 1) ∈ F q n t x , with gcd ⁡ (s , n) = 1. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a q n -polynomial. Finally, as an application of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in PG (1 , q 6) presented in [9]. [ABSTRACT FROM AUTHOR]

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two: (a) provide a characterization for the inclusion relation between the closures (in the standard topology) of bundles; and (b) are the bundles open in their closure? The main goal of this paper is providing an explicit answer to these two questions. In order to get this answer, we also review and/or formalize some notions and results already existing in the literature. We also prove that bundles of matrices under similarity, as well as bundles of matrix polynomials (defined as the set of m × n matrix polynomials of the same grade having the same spectral information, up to the eigenvalues) are open in their closure. [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]

We consider two algorithms that use the block Macaulay matrix to solve (rectangular) multiparameter eigenvalue problems (MEPs). On the one hand, a multidimensional realization problem in the null space of the block Macaulay matrix constructed from the coefficient matrices of an MEP results in a standard eigenvalue problem (SEP), the eigenvalues and eigenvectors of which yield the solutions of that MEP. On the other hand, we propose a complementary algorithm to solve MEPs that considers the data in the column space of the sparse and structured block Macaulay matrix directly, avoiding the computation of a numerical basis matrix of the null space. This paper generalizes, in a certain sense, traditional Macaulay matrix techniques from multivariate polynomial system solving to the block Macaulay matrix in the MEP setting. [ABSTRACT FROM AUTHOR]

This paper studies generic and perturbation properties inside the linear space of m × (m + n) polynomial matrices whose rows have degrees bounded by a given list d 1 , ... , d m of natural numbers, which in the particular case d 1 = ⋯ = d m = d is just the set of m × (m + n) polynomial matrices with degree at most d. Thus, the results in this paper extend to a much more general setting the results recently obtained in [29] only for polynomial matrices with degree at most d. Surprisingly, most of the properties proved in [29] , as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d 1 , ... , d m , and with right minimal indices differing at most by one and having a sum equal to ∑ i = 1 m d i , and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly. [ABSTRACT FROM AUTHOR]

A real square matrix is algebraically positive if there exists a real polynomial f such that f (A) is a positive matrix. In this paper, we give a sufficient condition for a sign pattern matrix to allow algebraic positivity, and give some methods to construct higher-order algebraically positive matrices from some lower-order algebraically positive matrices. We also propose two conjectures related to the problem of allowing algebraic positivity. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the problem of locating the zeros of regular polynomials of a quaternionic variable with restricted quaternionic coefficients. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results generalize some recently proved results about the distribution of zeros of a quaternionic polynomial. [ABSTRACT FROM AUTHOR]

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concepts. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the Hadamard matrix product. Thereby, some standard results arising for Max-Times systems with nonnegative entries appear as a special case. The case of two sided systems is also analyzed. In addition, a notion of eigenvalue in limit is considered. It is shown that one can construct a special semi-continuous regularized polynomial whose zeros are related to the eigenvalues of a matrix with nonnegative entries. [ABSTRACT FROM AUTHOR]

The location of the roots of a quadratic scalar polynomial may be identified from its coefficients. This paper shows that when the coefficients of the polynomial are square matrices, then appropriate generalizations of some of these statements hold for the eigenvalues of the resulting quadratic matrix polynomial. The locations of the eigenvalues are described with respect to the imaginary axis, the unit circle or the real line. The results lead to upper bounds on some important distances associated with quadratic matrix polynomials. The principal tool used is an eigenvalue localization technique using block Geršgorin sets applied to certain linearizations of these polynomials that come from well known vector spaces. New bounds on the eigenvalues of the matrix polynomial arising from these localizations are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a general formula for the characteristic polynomial of a finitely dimensional representation of Lie algebra sl (2 , C) and the form for these characteristic polynomials, and prove there is one to one correspondence between representations and their characteristic polynomials. Moreover we define a monoid structure on these characteristic polynomials. [ABSTRACT FROM AUTHOR]

As is well known, any complex cyclic matrix A is similar to the unique companion matrix associated with the minimal polynomial of A. On the other hand, a cyclic matrix over a division ring F is similar to a companion matrix of a polynomial which is defined up to polynomial similarity. In this paper we study more rigid canonical forms by embedding a given cyclic matrix over a division ring F into a controllable or an observable pair. Using the characterization of ideals in F [ z ] in terms of controllable and observable pairs we consider ideal interpolation schemes in F [ z ] which merge into polynomial interpolation problems containing both left and right interpolation conditions. [ABSTRACT FROM AUTHOR]

Instability in Jacobians is determined by the presence of an eigenvalue lying in the right half plane. The coefficients of the characteristic polynomial contain information related to the specific matrix elements that play a greater destabilising role. Yet the destabilising circuits, or cycles, constructed by multiplying these elements together, form only a subset of all the cycles comprising a given system. This paper looks at the destabilising cycles in three sign-restricted forms in terms of sets of the matrix elements to explore how sign structure affects how the elements contribute to instability. This leads to quite rich combinatorial structure among the destabilising cycle sets as set size grows within the coefficients of the characteristic polynomial. [ABSTRACT FROM AUTHOR]

Given a truncated sequence of positive numbers a = (a j) j = 0 m , the subnormal completion problem (originally posed and geometrically solved by J. Stampfli) asks whether or not there exists a subnormal weighted shift operator on ℓ 2 whose initial weights are given by a. Subsequently a concrete solution based on a solution of the truncated Stieltjes moment problem was discovered by Curto and Fialkow. In this paper we will consider a matricial analogue of the subnormal completion problem, where the truncated sequence of positive numbers (a j) j = 0 m is replaced by a truncated sequence of positive definite matrices. We will provide concrete conditions for a solution based on the parity of m and make a connection with a matricial truncated Stieltjes moment problem. We will also put forward a certain canonical subnormal completion which is minimal in the sense of the norm of the corresponding weighted shift operator and describe the support of the corresponding matricial Berger measure in terms of the zeros of a matrix polynomial which describes the initial positive rank preserving extension. [ABSTRACT FROM AUTHOR]

The tropical semiring R ∪ { − ∞ } is the semiring with addition "max" and multiplication "+". Tropical quadratic forms are represented by tropical symmetric matrices. Tropical quadratic forms with three variables define tropical projective quadratic plane curves. In this paper, we characterize tropical projective quadratic plane curves in terms of the eigenvalue problem for tropical matrices. In particular, we focus on the curved part of a tropical projective quadratic plane curve, that is, the cell never contained in any tropical projective line. We first prove that algebraic eigenvalues, i.e., roots of the characteristic polynomial, of a matrix express the minimum distance from the origin to the curved part. We then show that an algebraic eigenvector with respect to the minimum algebraic eigenvalue of a matrix indicates the direction to the nearest point in the curved part from the origin. [ABSTRACT FROM AUTHOR]

In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over F q of degree at most k , with 2 ≤ k < q , which pairwise take the same value on at least one point, consist of all polynomials f of degree at most k such that f (x) = y for some fixed x and y in F q. We also discuss this problem for ovoidal Minkowski planes, and we investigate the largest families of circles pairwise intersecting in two points in circle geometries. [ABSTRACT FROM AUTHOR]

Abstract We construct a new family of strong linearizations of rational matrices considering the polynomial part of them expressed in a basis that satisfies a three term recurrence relation. For this purpose, we combine the theory developed by Amparan et al. (2018) [4] , and the new linearizations of polynomial matrices introduced by Faßbender and Saltenberger (2017) [15]. In addition, we present a detailed study of how to recover eigenvectors of a rational matrix from those of its linearizations in this family. We complete the paper by discussing how to extend the results when the polynomial part is expressed in other bases, and by presenting strong linearizations that preserve the structure of symmetric or Hermitian rational matrices. A conclusion of this work is that the combination of the results in this paper with those in Amparan et al. (2018) [4] , allows us to use essentially all the strong linearizations of polynomial matrices developed in the last fifteen years to construct strong linearizations of any rational matrix by expressing such a matrix in terms of its polynomial and strictly proper parts. [ABSTRACT FROM AUTHOR]

In this paper, we consider structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, ⋆-even and ⋆-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases. [ABSTRACT FROM AUTHOR]

For a (finite or infinite dimensional) vector space V , the notion of a symmetric Jordan canonical form of an operator T ∈ L ( V ) having a minimal polynomial is defined and used to verify the relation between the notions of “Jordan canonical form” and “rational canonical form.” The paper extends and repairs Theorem 2.2 of Radjabalipour (2013) [6] . In particular, it is shown that there exists an auxiliary nilpotent operator S ∈ L ( W ) , depending on T , such that every Jordan canonical form of S yields a symmetric Jordan canonical form and, if the characteristic of the ground field is zero, a rational canonical form for T . The paper concludes with a direct proof of the symmetric Jordan canonical form which “integrates” into a rational canonical form. [ABSTRACT FROM AUTHOR]

A celebrated result of Karpelevič describes Θ n , the collection of all eigenvalues arising from the stochastic matrices of order n. The boundary of Θ n consists of roots of certain one-parameter families of polynomials, and those polynomials are naturally associated with the so-called reduced Ito polynomials of Types 0, I, II and III. In this paper we explicitly characterise all n × n stochastic matrices whose characteristic polynomials are of Type 0 or Type I, and all sparsest stochastic matrices of order n whose characteristic polynomials are of Type II or Type III. The results provide insights into the structure of stochastic matrices having extreme eigenvalues. [ABSTRACT FROM AUTHOR]

In 1972, Cvetković proved that if G is an n -vertex simple graph with the chromatic number k , then its spectral radius is at most the spectral radius of the n -vertex balanced complete k -partite graph. In this paper, we analyze the characteristic polynomial of a digraph D to prove a tight upper bound for the spectral radius of D in terms of the number of vertices and the chromatic number of D ; we also characterize when equality holds. This provides a simple proof of a result by Lin and Shu [6]. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices and m edges (i.e. an (n , m) -graph) and L (G) be the Laplacian matrix of G. The Laplacian characteristic polynomial of G is defined as P (G ; λ) = det ⁡ (λ I − L (G)) = ∑ i = 0 n (− 1) i c i (G) λ n − i , where c i (G) is referred as the i -th Laplacian coefficient of G. Denote G n , m by the set of all connected (n , m) -graphs. A connected graph H ∈ G n , m is called c i -minimal if c i (H) ≤ c i (G) holds for each G ∈ G n , m and is called uniformly minimal if H is c i -minimal for i = 0 , 1 , ... , n. In this paper, we prove that each c i -minimal graph in G n , m is a threshold graph for 2 ≤ i ≤ n − 2. Moreover, we prove that there does not exist uniformly minimal graphs in G n , n + 3 , n ≥ 6. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field with q = p λ elements. Maximum Rank metric codes (MRD for short) are subsets of M m × n (F q) whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2015) presented a new class of MRD codes over F q called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes H k , s (L 1 , L 2). The equivalence and duality of twisted Gabidulin codes were discussed by Lunardon, Trombetti, and Zhou (2016). A new class of MRD codes in M 2 n × 2 n (F q) was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case L 1 (x) = x , where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of H k , s (x , L (x)) and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of some twisted Gabidulin codes. [ABSTRACT FROM AUTHOR]

In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of η -Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of η -Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define n + 1 types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss periodicity. We show that the mixed paths are periodic for any η. In addition, we provide a necessary and sufficient condition for a mixed cycle to be periodic and determine their periods. [ABSTRACT FROM AUTHOR]

We study when functions of the eigenvalues of the pencil (1) Re (e − i t A) = cos ⁡ (t) Re A + sin ⁡ (t) Im A are constant functions of t. The results are then applied to questions regarding the numerical range, the higher rank numerical range and the c -numerical range, and we derive trace type conditions for when these numerical ranges are disks centered at 0. The theory of symmetric polynomials plays an important part in the proofs. [ABSTRACT FROM AUTHOR]

Let G be a connected simple graph with characteristic polynomial P G (x). The irreducibility of P G (x) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P 2 n + 1 − 2 (n ≥ 1) to certain vertices; union and join K 1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results. [ABSTRACT FROM AUTHOR]