Our paper is devoted to the investigations of doubly alternative zero divisors of the real Cayley–Dickson split-algebras. We describe their annihilators and orthogonalizers and also establish the ...relationship between centralizers and orthogonalizers for such elements. Then we obtain an analogue of the real Jordan normal form in the case of the split-octonions. Finally, we describe commutativity, orthogonality, and zero divisor graphs of the split-complex numbers, the split-quaternions, and the split-octonions in terms of their diameters and cliques.
Let F be an algebraically closed field and Mn be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with Mn as vertices, and two distinct matrices A,B are adjacent ...if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of Mn preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T1,T2:Mn→Mn satisfying the condition A+λB is singular if and only if T1(A)+λT2(B) is singular, for any A,B∈Mn and any non-zero λ∈F. Namely, we prove that in this case T1=T2 and they are of the form T1(A)=T2(A)=PAQ for all A∈Mn, or of the form T1(A)=T2(A)=PAtQ for all A∈Mn, where P,Q∈Mn are non-singular matrices.
Let Mn(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:Mn(F)→Mn(F) is additive if f(A+B)=f(A)+f(B) for all A,B∈Mn(F). The ...main goal of this paper is to prove that for n>2 there are no additive surjective maps T:Mn(F)→Mn(F) such that per(T(A))=det(A) for all A∈Mn(F). Also we show that an arbitrary additive surjective map T:Mn(F)→Mn(F) which preserves permanent is linear and thus can be completely characterized.
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 2n−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 2n can be realized. We also investigate the set of integer values that the permanent function cannot attain on the set of (0,1)-matrices.
Relation Graphs of the Sedenion Algebra Guterman, A. E.; Zhilina, S. A.
Journal of mathematical sciences (New York, N.Y.),
06/2021, Volume:
255, Issue:
3
Journal Article
Peer reviewed
Let 𝕊 denote the algebra of sedenions and let Γ
O
(𝕊) denote its orthogonality graph. One can observe that every pair of zero divisors in 𝕊 generates a double hexagon in Γ
O
(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 Γ
O
(𝕊) is described, and its diameter is found. Then, the bijection between the connected components of Γ
O
(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.
Let Qn denote the space of all n×n skew-symmetric matrices over the complex field C and T:Qn→Qn be a map satisfying the condition dχ′(T(A)+zT(B))=dχ(A+zB) for all matrices A,B∈Qn and all constants ...z∈C. Here χ and χ′ are irreducible characters of the permutation group Sn and dχ(C) denotes the immanant of the matrix C associated with the character χ. Let Pn 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 Pn. The second is to characterize such maps T if χ and χ′ are proportional on Pn. In particular, we prove that T is bijective and linear. Observe that the general problem of the characterization for bijective linear maps on Qn, 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.
An Upper Bound for the Chainable Index Alpin, Yu. A.; Guterman, A. E.; Shafeev, E. R.
Journal of mathematical sciences,
05/2023, Volume:
272, Issue:
4
Journal Article
Peer reviewed
Open access
The paper considers the chainable index of a square matrix of order n and proves that it does not exceed
n
− 1. Also it is demonstrated that every integer in between 0 and
n
− 1 is a value of the ...chainable index.
The lengths of two particular cases of (possibly non-unital) composition algebras, namely standard composition algebras and Okubo algebras over an arbitrary field, are computed. These results finish ...the complete description of lengths of symmetric composition algebras and finite-dimensional flexible composition algebras.
Let Matn(F) denote the set of square n×n matrices over a field F of characteristic different from two. The permanental rank prk(A) of a matrix A∈Matn(F) is the size of the maximal square submatrix in ...A with nonzero permanent. By Λk and Λ≤k we denote the subsets of matrices A∈Matn(F) with prk(A)=k and prk(A)≤k, respectively. In this paper for each 1≤k≤n−1 we obtain a complete characterization of linear maps T:Matn(F)→Matn(F) satisfying T(Λ≤k)=Λ≤k or bijective linear maps satisfying T(Λ≤k)⊆Λ≤k. Moreover, we show that if F is an infinite field, then Λk is Zariski dense in Λ≤k and apply this to describe such bijective linear maps satisfying T(Λk)⊆Λk.
