Split‐type octonion matrix

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form , where e_m's have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts.     

On a Special Case of the Herbert Stahl Theorem

The BMV conjecture states that for n ×  n Hermitian matrices A and B the function f_A,B(t) = trace e^tA+B is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for the special case rank A = 1. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.     

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B _n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB _n contains an ideal K _n of dimension (2 ⁿ  ? 1)², and we construct an explicit isomorphism of K _n with the matrix algebra M _{2 ⁿ ?1}(F).     

Maps preserving square-zero matrices on the algebra consisting of all upper triangular matrices

Let F be a field, T _n (F) (respectively, N _n (F)) the matrix algebra consisting of all n × n upper triangular matrices (respectively, strictly upper triangular matrices) over F. A ∈ T _n (F) is said to be square zero if A ² = 0. In this article, we firstly characterize non-singular linear maps on N _n (F) preserving square-zero matrices in both directions, then by using it we determine non-singular linear maps on T _n (F) preserving square-zero matrices in both directions.     

Localisations of non prime e-vh-orders

Abstract

If A is an n × n complex matrix and x ∈ ? ⁿ , the conjecture is that if we take the kth power of each component of Ax, the resulting vector belongs to the range of the matrix obtained by taking the kth power of the entries of AA ^?, where A ^? is the adjoint of A. The conjecture is proved here for any k ≥ 2 when we add assumptions of either low dimension (namely, n ≤ 4) or low corank (0, 1, and, with some technical restrictions, 2). This problem arises in the study of the Jacobian Conjecture.     

On Strongly J-Clean Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of T _n (R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation.     

On reconstruction of a matrix by its minors

In this paper, we reconstruct matrices from their minors, and give explicit formulas for the reconstruction of matrices of orders 2 × 3, 2 × 4, 2 × n, 3 × 6 and m × n. We also formulate the Plücker relations, which are the conditions of the existence of a matrix related to its given minors.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

The difference between doubly nonnegative and completely positive matrices

The convex cone of n×n completely positive (CP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for n4 only, every DNN matrix is CP. In this paper, we investigate the difference between 5×5 DNN and CP matrices. Defining a bad matrix to be one which is DNN but not CP, we: (i) design a finite procedure to decompose any n×n DNN matrix into the sum of a CP matrix and a bad matrix, which itself cannot be further decomposed; (ii) show that every bad 5×5 DNN matrix is the sum of a CP matrix and a single bad extreme matrix; and (iii) demonstrate how to separate bad extreme matrices from the cone of 5×5 CP matrices.     

Tournament matrices and their generalizations,I.

If M is any complex matrix with rank (M + M ^* + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n ? 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + M^T + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n? {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.     

On the optimal computation of a set of symmetric and persymmetric bilinear forms

A special class $\text{T}$ⁿ of n×n matrices is described, which has tensor rank n over the real field. A tensor base for general symmetric, persymmetric, both symmetric and persymmetric matrices and Toeplitz symmetric matrices can be defined in terms of the tensor bases of $\text{T}$^l for some different values of l. It is proved that both symmetric and persymmetric n×n matrices and Toeplitz symmetric n×n matrices have tensor rank $\text{[}\text{(n+1)}{}^2\text{4}\text{]}$ and 2n?2, respectively, in the real field.     

On semidefinite linear complementarity problems

Given a linear transformation L:? ⁿ →? ⁿ and a matrix Q∈? ⁿ , where ? ⁿ is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,? ⁿ ₊,Q) over the cone ? ⁿ ₊ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R ^n×n , we consider the linear transformation L _A :? ⁿ →? ⁿ defined by L _A (X):=AX+XA ^T and show that the P- and Q-properties for L _A are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov. Received: March 1999 / Accepted: November 1999?Published online April 20, 2000     

On a conjecture about the eigenvalues of doubly stochastic matrices

According to a long standing conjecture, the geometric location of eigenvalues of doubly stochastic matrices of order n is exactly the union of regular k-gons anchored at 1 in the unit disc for 2 ≤ k ≤ n. It is easy to verify this fact for n?=?2,?3. But, for n?≥?4, it has been an open question. We show that this conjecture is wrong for n?=?5.     

Adjacency preserving bijective maps on triangular matrices over any division ring

Let 𝕋 _n (D) be the set of n × n upper triangular matrices over a division ring D. We characterize the adjacency preserving bijective maps in both directions on 𝕋 _n (D) (n ≥ 3). As applications, we describe the ring semi-automorphisms and the Jordan automorphisms on upper triangular matrices over a simple Artinian ring.     

Products of involutory matrices. I

It is shown that, for every integer ?1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n?4 or F has prime order ?5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

The complete matrix ring over an adjoint regular ring is adjoint regular

We prove that the ring of all n×n matrices over an adjoint regular ring is adjoint regular, thus confirming a longstanding conjecture in the theory of adjoint semigroups.     

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Let R ? ? be a GCD-domain. In this article, Weinberg's conjecture on the n × n matrix algebra M _n (R) (n ≥ 2) is proved. Moreover, all the lattice orders (up to isomorphisms) on a full 2 × 2 matrix algebra over R are obtained.     

Modular sequences and modular Hadamard matrices

For every n divisible by 4, we construct a square matrix H of size n, with coefficients ± 1, such that H · H^t ≡ nI mod 32. This solves the 32‐modular version of the classical Hadamard conjecture. We also determine the set of lengths of 16‐modular Golay sequences. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 187–214, 2001     

Discriminant of symmetric matrices as a sum of squares and the orthogonal group

It is proved that the discriminant of n × n real symmetric matrices can be written as a sum of squares, where the number of summands equals the dimension of the space of n‐variable spherical harmonics of degree n. The representation theory of the orthogonal group is applied to express the discriminant of 3 × 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares. It is proved that the discriminant of 4 × 4 real symmetric matrices can be written as a sum of seven squares. These improve results of Kummer from 1843 and Borchardt from 1846. © 2010 Wiley Periodicals, Inc.     

A technique for constructing non‐embeddable quasi‐residual designs

We propose a technique for constructing two infinite families of non‐embeddable quasi‐residual designs as soon as one such design satisfying certain conditions exists. The main tools are generalized Hadamard matrices and balanced generalized weighing matrices. Starting with a specific non‐embeddable quasi‐residual 2‐(27,9,4) design, we construct for every positive integer m a non‐embeddable 2‐(3^m,3^m?1,(3^m?1?1)/2)‐design, and, if r_m=(3^m?1)/2 is a prime power, we construct for every positive integer n a non‐embeddable design. For each design in these families, a symmetric design with the corresponding parameters is known to exist. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 160–172, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.900