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Let C(A)=A∘A−T{\mathcal{C}}\left(A)=A\circ {A}^{-T} be the combined matrix of an invertible matrix AA, where ∘\circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an HH-matrix whose comparison matrix is singular. In particular, we focus on C(A){\mathcal{C}}\left(A) when AA is diagonally equipotent, and we study whether C(A){\mathcal{C}}\left(A) is an HH-matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.

Subspace codes and constant dimension codes have become a widely investigated research topic due to their significance to error control in random linear network coding. Rank metric codes in Ferrers diagrams can be used to construct good subspace codes and constant dimension codes. In this paper, three constructions of Ferrers diagram rank metric codes are presented. The first two constructions are based on subcodes of maximum rank distance codes, and the last one generates new codes from known Ferrers diagram rank metric codes. Each of these constructions produces optimal codes with different diagrams and parameters for which no optimal construction was known before. [ABSTRACT FROM AUTHOR]

We prove that the commutative nearly absolute valued division algebras have dimension ≤2. This extends a well-known theorem by H. Hopf asserting the same property for each commutative real division algebra of finite dimension. As an application we prove that R , C and C ⋆ are the unique commutative real nearly absolute valued division algebras with norm satisfying ‖ x 2 ‖ = ‖ x ‖ 2 . [ABSTRACT FROM AUTHOR]

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of monic Hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representation (MDR) of size 2 if it exists. It is known that a quadratic polynomial f ( x ) = x T A x + b T x + 1 has a symmetric MDR of size n + 1 if A is negative semidefinite . We prove that if a quadratic polynomial f ( x ) with A which is not negative semidefinite has an MDR of size greater than 2, then it has an MDR of size 2 too. Finally, we characterize all quadratic polynomials that exhibit MDRs of any size. [ABSTRACT FROM AUTHOR]

Consider T ( F)-the ring of all n × n upper triangular matrices defined over some field F. A map φ is called a zero product preserver on T ( F) in both directions if for all x, y ∈ T ( F) the condition xy = 0 is satisfied if and only if φ( x) φ( y) = 0. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map φ may act in any bijective way, whereas for the zero divisors and zero matrix one can write φ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix x into a matrix x′ such that { y ∈ T ( F): xy = 0} = { y ∈ T ( F): x′ y = 0}, { y ∈ T ( F): yx = 0} = { y ∈ T ( F): yx′ = 0}. [ABSTRACT FROM AUTHOR]

Considering the natural topology or S -density topology on the domain and on the range we obtain different families of continuous functions f : R → R . We prove that certain differences of those families are strongly c -algebrable, i.e., they contain c -generated free subalgebras. In particular, we obtain strengthenings of recent results of J. Hejduk, A. Loranty, R. Wiertelak, S. Lindner and M. Terepeta. [ABSTRACT FROM AUTHOR]

We construct the definition of allowable graphs of the nonlinear Schrödinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving 3-dimensional blocks of the normal form of the nonlinear Schrödinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both n th and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested sums of their variable coefficients. It makes explicit all the sum of products involved in the previous representations of such solutions. Some basic applications are also illustrated. [ABSTRACT FROM AUTHOR]

A class of m -by- n real (complex) matrices is semi-algebraic (SA) if membership in it may be checked via a finite list of polynomial inequalities in the entries (the real and imaginary parts of the entries). Real algebraic geometry is used to show that (1) very many familiar classes of matrices (some perhaps surprising) are SA and (2) that two important derived problems (matrix completions and Hadamard Duals) lead to an SA solution when they come from an SA class. This means that at least there exists a finite solution to these problems though it may be difficult to find. The methodology likely extends to other problems of interest. [ABSTRACT FROM AUTHOR]

A set of vertices in a graph is called independent if no two vertices of the set are connected by an edge. In this paper we use the state matrix recursion algorithm, developed by Oh, to enumerate independent vertex sets in a grid graph and even further to provide the generating function with respect to the number of vertices. We also enumerate bipartite independent vertex sets in a grid graph. The asymptotic behavior of their growth rates is presented. [ABSTRACT FROM AUTHOR]

We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes first proved by Delsarte employing association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, MRD codes). In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive both a recursive and an explicit formula for the number of $$k \times m$$ matrices over a finite field with given rank and $$h$$ -trace. [ABSTRACT FROM AUTHOR]

We consider trace functions ( A , B ) ↦ Tr [ ( A q / 2 B p A q / 2 ) s ] where A and B are positive n × n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of A q / 2 B p A q / 2 and convexity/concavity of the closely related trace functional Tr [ A q / 2 B p A q / 2 C r ] . The concavity questions are completely resolved, thereby settling cases left open by Hiai; the convexity questions are settled in many cases. As a consequence, the Audenaert–Datta Rényi entropy conjectures are proved for some cases. [ABSTRACT FROM AUTHOR]

We prove that a nonsymmetric normal entry pattern of order n ( n ≥ 3 ) has at most n ( n − 3 ) / 2 + 3 distinct indeterminates and up to permutation similarity this number is attained by a unique pattern which is explicitly described. An open problem is posed. [ABSTRACT FROM AUTHOR]

The Ducci map is defined by taking a vector [ v 1 , … , v n ] T ∈ R n to [ | v 1 − v 2 | , … , | v n − v 1 | ] T . We concern ourselves with the Ducci map's action in R 3 , establishing a connection between the sequences of matrices associated with the action of the Ducci map, continued fraction representations of the real numbers, and the Stern–Brocot tree. It is shown that the real numbers have a representation via sequences of Ducci matrices, and in this Ducci number system there are essentially three types of matrix sequences: one type corresponding to the rationals, one to the quadratic irrationals, and then those types that correspond to all other real numbers. This mirrors the situation for continued fraction representations of the reals. It follows that the Ducci map on R 3 is closely connected to the Euclidean algorithm and, through its action, locates best rational approximations to the irrationals. [ABSTRACT FROM AUTHOR]

Let T ( n , F ) be the set of all n × n upper triangular matrices over a field F . Recently, by using a technique from graph theory, L. Wang in [8] proved that a bijective mapping σ on T ( n , F ) , with F a finite field, preserves zero product, i.e., σ ( x ) σ ( y ) = 0 whenever x y = 0 , if and only if σ can be decomposed into the product of an inner automorphism, an automorphism of T ( n , F ) induced by an automorphism of F and a regular automorphism of the zero-divisor graph of T ( n , F ) . Algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra. This viewpoint encourages us to determine nonlinear mappings χ on T ( n , F ) which are derivable at zero point, i.e., χ ( x ) y + x χ ( y ) = 0 whenever x y = 0 . It is proved that a mapping χ on T ( n , F ) is derivable at zero point if and only if χ is the sum of an inner derivation, an additive derivation of T ( n , F ) induced by an additive derivation of F , and a regular mapping on T ( n , F ) . [ABSTRACT FROM AUTHOR]

Here a known result on the structure of finite Hessenberg matrices is extended to infinite Hessenberg matrices. Its consequences for the example of infinite Hessenberg–Toeplitz matrices are described. The results are applied also to the inversion of infinite tridiagonal matrices via recurrence relations. Moreover, since there are available free parameters, different inverses can be associated with a given invertible tridiagonal matrix. [ABSTRACT FROM AUTHOR]

Let N = N( R) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R. A map φ on N is called preserving commutativity in both directions if xy = yx ⇔ φ( x) φ( y) = φ( y) φ( x). In this paper, we prove that each invertible linear map on N preserving commutativity in both directions is exactly a quasi-automorphism of N, and a quasi-automorphism of N can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings. [ABSTRACT FROM AUTHOR]

Abstract: We address the issue of simplifying symbolic polynomials on non-commutative variables. The problem is motivated by applications in optimization and various problems in systems and control. We develop theory for polynomials which are linear in a subset of the variables and develop algorithms to produce representations which have the minimal possible number of terms. The results can handle polynomial matrices as well as block-matrix variables. [Copyright &y& Elsevier]

Abstract: A real matrix A is called sign-central if has a nonzero nonnegative solution x for every matrix with the same sign pattern as A. A sign-central matrix A is called tight sign-central if the Hadamard(entrywise) product of any two columns of A contains a negative component. Hwang et al. [S.G. Hwang, I.P. Kim, S.J. Kim, X.D. Zhang, Tight sign-central matrices, Linear Algebra Appl. 371 (2003) 225–240] proved that, for a positive integer m, there exists an m × n (0,1,−1) tight sign-central matrix A with no zero rows if and only if m +1⩽ n ⩽2 m . They also determined the lower bound of the number of columns of a tight sign-central matrix with no zero rows in terms of the number of rows and the number of zero entries of the matrix along with the characterization of the equality case. For an m × n matrix A, the sparsity of A is the ratio σ(A)/mn where σ(A) denotes the number of zero entries of A. In this paper, we determine the maximum number and the minimum number of zero entries of an m × n tight sign-central matrix with no zero rows for each pair (m, n) of positive integers with m +1⩽n⩽2 m . We also determine the maximum sparsity of tight sign-central matrices with m nonzero rows in terms of m for each positive integer m. [Copyright &y& Elsevier]

Abstract: Let ∥·∥ be a matrix norm on and let be a finite set of matrices in . We define to be the maximum norm of a product of n elements of . We show that there is a gap in the possible growth of , showing that grows either at least exponentially or is bounded by a polynomial in n of degree at most d−1. Moreover, we show that the growth is bounded by a polynomial if and only if every element of the semigroup generated by has all of its eigenvalues on or inside the unit circle. [Copyright &y& Elsevier]

Abstract: A description is given of those ray-patterns, which will be called inverse closed ray-nonsingular, of complex matrices that contain invertible matrices only and are closed under inversion. Here, two n×n matrices are said to belong to the same ray-pattern if in each position either the entries of both matrices are zeros, or both entries are nonzero and their quotient is positive. Possible Jordan forms of matrices in the inverse closed ray-nonsingular ray-patterns are characterized. [Copyright &y& Elsevier]

Abstract: It is proved that in two-dimensional unital real algebras a step-by-step construction of minimal convex invertible sets terminates in two steps. An analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps, and the complex field is the only finite dimensional complex algebra with this property. Connections are made with the matrix sign function. The matrix sign function of a matrix belongs to the convex invertible set generated by the matrix. An example is given to show that the matrix sign function cannot always be obtained in two steps, by following the step-by-step construction. [Copyright &y& Elsevier]

Abstract: Let D be a division ring with an involution . Assume that is a proper subfield of D and is contained in the center of D. Let be the set of n×n skew-Hermitian matrices over D. If and rank(H1−H2)=1, H1 and H2 are said to be adjacent. The fundamental theorem of the geometry of skew-Hermitian matrices over D is proved: Let n⩾2 and be a bijective map of to itself, which preserves the adjacency. Then is of the form , where α∈F*, P∈GLn(D), , and σ is an automorphism of D. [Copyright &y& Elsevier]

Abstract: Using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense. As an application some consequences of this property in the area of time-varying stability radii are discussed. In particular, using the implicit function theorem sufficient conditions for Lipschitz continuity are derived. An example is presented of a linearly increasing family of matrix polytopes for which the proximal subgradient of the generalized spectral radius at a certain polytope contains 0, so that the implicit function theorem is not applicable in all cases. [Copyright &y& Elsevier]

In this paper we work with pairs of zero–one matrices whose product is the full upper-triangular zero–one matrix. We establish a lower bound for the sum of the non-zero entries in them, using elementary concepts of linear algebra. With this algebraic method we reach the best possible lower bound, since pairs of such matrices may be found whose amount of non-zero entries is exactly our lower bound. [Copyright &y& Elsevier]