Search

We suggest a new method which allows us to compute the lengths of (possibly non-unital) standard composition algebras over an arbitrary field F with char F≠2.

In this paper we study the relations between numerical characteristics of finite dimensional algebras and such classical combinatorial objects as additive chains. We study the behavior of the length function via so-called characteristic sequences of quadratic algebras. As one of our main results, we prove the sharp upper bound for the length of quadratic algebras in terms of the Fibonacci numbers depending on the dimension of the algebra. Moreover, we show that quadratic algebras have the extremal behavior with respect to this bound. In addition, we obtain the description of the set of characteristic sequences for quadratic algebras. Namely, we completely determine the set of combinatorial properties which are satisfied for characteristic sequences of quadratic algebras and show that they belong to the family of additive chains known in combinatorics. Conversely, for a given integer sequence being an additive chain and satisfying these combinatorial properties, we construct a quadratic algebra with a characteristic sequence equal to this sequence. The obtained information on characteristic sequences is then applied to investigate the problem of realizability for the length function. In particular, we determine certain subsets of integers which are not realizable as values of the length function on quadratic algebras.

Let Q n denote the space of all n × n skew-symmetric matrices over the complex field C and T : Q n → Q n be a map satisfying the condition d χ ′ ( T ( A ) + z T ( B ) ) = d χ ( A + z B ) for all matrices A , B ∈ Q n and all constants z ∈ C . Here χ and χ ′ are irreducible characters of the permutation group S n and d χ ( C ) denotes the immanant of the matrix C associated with the character χ. Let P n be the set of permutations with no cycles of odd length in the decomposition into the product of independent cycles. The main goal of this paper is twofold. The first one is to show that there are no maps T satisfying the above conditions if n ≥ 6 is even and χ and χ ′ are not proportional on P n . The second is to characterize such maps T if χ and χ ′ are proportional on P n . In particular, we prove that T is bijective and linear. Observe that the general problem of the characterization for bijective linear maps on Q n , that convert one immanant into another, remained an open problem with the exception for determinant and permanent. Our results include all known characterizations for immanant preserving and converting maps and provide the complete solution of this problem.

Let Qn denote the space of all n × n skew-symmetric matrices over the complex field ℂ. It is proved that for n = 4, there are no linear maps T : Q4 → Q4 satisfying the condition dχ' (T (A)) = dχ(A) for all matrices A ∈ Q4, where χ, χ' ∈ {1, ∈, [2, 2]} are two distinct irreducible characters of S4. In the case χ = χ' = 1, a complete characterization of the linear maps T : Q4 → Q4 preserving the permanent is obtained. This case is the only one corresponding to equal characters and remaining uninvestigated so far.

Let 𝕊 denote the algebra of sedenions and let ΓO(𝕊) denote its orthogonality graph. One can observe that every pair of zero divisors in 𝕊 generates a double hexagon in ΓO(𝕊). The set of vertices of a double hexagon can be extended to a basis of 𝕊 that has a convenient multiplication table. The set of vertices of an arbitrary connected component of ΓO(𝕊) is described, and its diameter is found. Then, the bijection between the connected components of ΓO(𝕊) and the lines in the imaginary part of the octonions is established. Finally, the commutativity graph of the sedenions is considered, and it is shown that all the elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies in between 3 and 4.

In the theory of matrix majorizations there are three types of majorizations that play an important role. These are weak, strong and directional majorizations for matrices. In this paper we characterize the linear operators converting each one of these majorizations to another one.

We combine matrix theory and graph theory methods to give a complete characterization of the surjective linear transformations of tropical matrices that preserve the cyclicity index. We show that there are non-surjective linear transformations that preserve the cyclicity index and we leave it open to characterize those.

Let M n ( F ) denote the set of square matrices of size n over a field F with characteristics different from two. We say that the map f : M n ( F ) → M n ( F ) is additive if f ( A + B ) = f ( A ) + f ( B ) for all A , B ∈ M n ( F ) . The main goal of this paper is to prove that for n > 2 there are no additive surjective maps T : M n ( F ) → M n ( F ) such that per ( T ( A ) ) = det ⁡ ( A ) for all A ∈ M n ( F ) . Also we show that an arbitrary additive surjective map T : M n ( F ) → M n ( F ) which preserves permanent is linear and thus can be completely characterized.

A full integral of a polynomial is defined as its integral with the property that any multiple root of the polynomial is a root of this integral. The paper investigates relationships between the existence of a full integral and the form of a polynomial. In particular, it is proved that a full integral exists if the polynomial has no more than one multiple root. On the other hand, if the number of multiple roots of a polynomial strictly exceeds the number of its simple roots increased by one, then the polynomial has no full integral. Bibliography: 7 titles.

Double occurrence words play an important part in genetics in describing epigenetic genome rearrangements. A useful geometric representation for double occurrence words is provided by the so-called assembly graphs. The paper investigates properties of the incidence matrices that correspond to the assembly graphs. An explicit matrix characterization of the simple assembly graphs of a given structure and a series of constructions, using these graphs and important for genetic investigations, are provided.

A lower and an upper bounds for the length of a direct sum of nonassociative algebras are obtained, and their sharpness is established. Note that while the lower bound for the length of a direct sum in the associative and nonassociative cases turns out to be the same, the upper bound in the nonassociative case significantly exceeds its associative counterpart.

In this paper, we provide the complete characterization of integer sequences that are characteristic sequences for general non-associative algebras, i.e., we determine the set of combinatorial prop...

We introduce the notion of length for non-associative finite-dimensional unitary algebras and obtain a sharp upper bound for the lengths of algebras belonging to this class. We also put forward a new method of characteristic sequences based on linear algebra technique, which provides an efficient tool for computing the length function in non-associative case. Then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra. In the last case the length is bounded in terms of the Fibonacci sequence. We present concrete examples demonstrating the sharpness of our bounds.

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B if Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.

In this paper we solve the problem of finding the length of group algebras of arbitrary finite abelian groups in the case when the characteristic of the ground field does not divide the order of the group. We show that these group algebras have maximal possible lengths for infinite fields and sufficiently large finite fields since they are one-generated. In case of small fields we prove that the length is bounded from above by a logarithmic function of the order of the group.

The notion of the scrambling index is a fundamental invariant in graph theory and in the theory of non-negative matrices and their applications. Namely, a scrambling index of a primitive directed g...

Let A be a (0, 1) -matrix such that P A is indecomposable for every permutation matrix P and there are 2 n + 3 positive entries in A . Assume that A is also nonconvertible in a sense that no change of signs of matrix entries, satisfies the condition that the permanent of A equals to the determinant of the changed matrix. We characterized all matrices with the above properties in terms of bipartite graphs. Here 2 n + 3 is known to be the smallest integer for which nonconvertible fully indecomposable matrices do exist. So, our result provides the complete characterization of extremal matrices in this class.

The paper evaluates the lengths of group algebras of all groups of orders not exceeding 7 over an arbitrary field.

The paper studies the anticommutativity condition for elements of arbitrary real Cayley–Dickson algebras. As a consequence, the anticommutativity graphs on equivalence classes of such algebras are classified. Under some additional assumptions on the algebras considered, an expression for the centralizer of an element in terms of its orthogonalizer is obtained. Conditions sufficient for this interrelation to hold are provided. Also examples of real Cayley–Dickson algebras in which the centralizer and orthogonalizer of an element are not interrelated in this way are considered.

Let Qn(ℂ) denote the space of all skew-symmetric n × n matrices over the complex field ℂ. The paper characterizes the linear mappings T : Qn(ℂ) → Qn(ℂ) that satisfy the condition per(T(A)) = per(A) for all matrices A ∈ Qn(ℂ) and an arbitrary n > 4.

For the pairs of quasi-commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators.

In this paper we investigate the permanent of ( − 1 , 1 ) -matrices over fields of zero characteristics and our main goal is to provide a sharp upper bound for the value of the permanent of such matrices depending on matrix rank, solving Wang's problem posed in 1974 by confirming Krauter conjecture formulated in 1985.

In this paper, we address the question of finding the length of group algebras of finite abelian groups in the case when the characteristic of the ground field divides the order of the group. We evaluate the exact length for an arbitrary abelian [Formula: see text]-group. For other groups we provide upper and lower bounds for the length of their group algebras.

Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.

It is shown that Birkhoff-James orthogonality knows everything about the smooth norms in reflexive Banach spaces and can also compute the dimensions of the underlying normed spaces., Comment: 18 pages and 1 figure

In the present paper, we introduce a general notion of quotient ring which is based on inverses along an element. We show that, on the one hand, this notion encompasses quotient rings constructed using various generalized inverses. On the other hand, such quotient rings can be viewed as Fountain–Gould quotient rings with respect to appropriate subsets. We also investigate the connection between partial order relations on a ring and on its ring of quotients.

We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are computed.

We study linear functions on the space of $n \times n$ matrices over a field which preserve or strongly preserve each of Green's equivalence relations ($\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$ and $\mathcal{J}$) and the corresponding pre-orders. For each of these relations we are able to completely describe all preservers over an algebraically closed field (or more generally, a field in which every polynomial of degree $n$ has a root), and all strong preservers and bijective preservers over any field. Over a general field, the non-zero $\mathcal{J}$-preservers are all bijective and coincide with the bijective rank-$1$ preservers, while the non-zero $\mathcal{H}$-preservers turn out to be exactly the invertibility preservers, which are known. The $\mathcal{L}$- and $\mathcal{R}$-preservers over a field with "few roots" seem harder to describe: we give a family of examples showing that they can be quite wild., Comment: 21 pages

We study the relation of mutual strong Birkhoff–James orthogonality in two classical $$C^*$$ -algebras: the $$C^*$$ -algebra $${\mathbb {B}}(H)$$ of all bounded linear operators on a complex Hilbert space H and the commutative, possibly nonunital, $$C^*$$ -algebra. With the help of the induced graph it is shown that this relation alone can characterize right invertible elements. Moreover, in the case of commutative unital $$C^*$$ -algebras, it can detect the existence of a point with a countable local basis in the corresponding compact Hausdorff space.

This paper deals with the important notion of majorization. We study majorization for matrices, and focus on ( 0 , 1 ) -matrices. We prove several results concerning such matrix majorization orders on the set of (0,1)-matrices, including characterizations for certain orders, and separate sufficient and necessary conditions for the so-called matrix majorization order. Some of these results are of combinatorial nature.

In this paper we discuss the values of the permanent of (0,1)-matrices of size n. Classical Brualdi–Newman theorem asserts that every integer value from 0 up to 2 n − 1 can be realized as the permanent of such a matrix. We obtain a result which is at least twice better and in particular we show that all nonnegative integer values which are less than or equal to 2 n can be realized. We also investigate the set of integer values that the permanent function cannot attain on the set of ( 0 , 1 ) -matrices.

We characterize linear transformations on matrices over commutative antinegative semirings that are monotone with respect to minus, star, and sharp partial orders.

The paper investigates the divisibility of the permanent function of (1,−1)-matrices by different powers of 2. It is shown that the Krauter–Seifter bound is the best possible one for generic (1,−1)-matrices.

Let matrices A,C ∈ Mn have eigenvalues α1, . . ., αn and γ1, . . . , γn, respectively. The set of complex numbers DC(A) = {det(A−UCU*) : U ∈ Mn, U*U = In} is called the C-determinantal range of A. The paper studies various conditions under which the relation DC(R S) = DC(S R) holds for some matrices R and S.

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring Mn(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring Mn(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.

In this paper we characterize those linear bijective maps on the monoid of all n × n square matrices over an anti-negative semifield (that is, a semifield which is not a field) which preserve each of Green's equivalence relations L , R , H , D , J and the corresponding four pre-orderings ≤ L , ≤ R , ≤ H , ≤ J . These results apply in particular to the tropical and boolean semirings.

Let M n ( F ) be the algebra of n × n matrices over the field F and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M n ( F ) cannot exceed 2 n − 2 . We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2 n − 2 is achieved.

In this paper we characterize surjective linear maps on matrices over antinegative semirings that preserve the set of matrices with maximal or minimal positive values of the scrambling index.

We describe matrices with extremal generalized centralizers over algebraically closed fields.

The concepts of differentiation and integration for matrices were introduced for studying zeros and critical points of complex polynomials. Any matrix is differentiable, however not all matrices are integrable. The purpose of this paper is to investigate the integrability property and characterize it within the class of diagonalizable matrices. In order to do this we study the relation between the spectrum of a diagonalizable matrix and its integrability and the diagonalizability of the integral. Finally, we apply our results to obtain a dual Schoenberg type inequality relating zeros of polynomials with their critical points.

Let Σ n (F) denote the space of all n×n symmetricmatrices over the complex field F, and χ be an irreducible character of S n and d χ the immanant associated with χ. The main objective of this paper is to prove that the maps Φ: Σ n (F) → Σ n (F) satisfying d χ(Φ(A) + αΦ(B)) = det(A + αB) for all singular matrices A, B ∈ Σ n (F) and all scalars α ∈ F are linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on the set of all symmetric matrices.

The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆). The length of ℝ as an algebra over itself is zero; the length of ℂ as an ℝ-algebra equals one. The purpose of the present paper is to prove that the lengths of the ℝ-algebras of quaternions and octonions equal two and three, respectively.