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Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we proved that each nonlinear non-global semi-Jordan triple higher derivable mapping on a 2-torsion free triangular algebra is an additive higher derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively. [ABSTRACT FROM AUTHOR]

This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by S ′ to compute Y S ′ X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems. [ABSTRACT FROM AUTHOR]

For general complex or real 1-parameter matrix flows A(t)n, n and for static matrices A ∈ ℂ n , n alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity Cn, n as A(t) = C− 1 ⋅ diag(A1(t),...,Aℓ(t)) ⋅ C or A = C− 1 ⋅diag(A1,...,Aℓ) ⋅ C with each diagonal block Ak(t) or Ak square and their number ℓ exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed 'eigenvectors' of one associated flow matrix B(ta) to find the coarsest simultaneous block structure for all flow matrices B(tb). The method works efficiently for all time-varying matrix flows A(t), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t, and likewise for all fixed entry matrices A. Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting 'divide and conquer' methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms. [ABSTRACT FROM AUTHOR]

The algorithm for solving the matrix equation Xn = A is well-known, where A is an given matrix and A is the unknown square matrix. This paper is concerned with the general case of the polynomial matrix equation , where. The study of the general case help us to solve any polynomial matrix equation. The main difficulty to solve the polynomial matrix equation is that, in general, the function is not invertible. Even if the function h is invertible, it is difficult to find the type of the inverse function and its derivatives. We designed an algorithm, which enables us to bypass anything related with the inverse function of h. In our algorithm we just used the polynomial function h and its derivatives. As it is easily understood, this is a very effective procedure and our algorithm can be used for every polynomial function h and any square matrix A. All the possible cases concerning the Jordan canonical form of the matrix A are examined and a formula of the matrix hoq(A) is given where q(x) is the interpolating polynomial to the data (4.1.1, 4.1.2). Mathematical types to calculate the number of different roots of the polynomial matrix equation and their algebraic multiplicity are also presented (5). We want to point out that the general case helps us to solve any polynomial matrix equation, which play an important role in many applications. [ABSTRACT FROM AUTHOR]

Let 풜 be a prime ring equipped with an involution ' * ' of order 2 and let m ≠ n be some fixed positive integers such that 풜 is 2 ⁢ m ⁢ n ⁢ (m + n) ⁢ | m - n | -torsion free. Let 풬 m ⁢ s ⁢ (풜) be the maximal symmetric ring of quotients of 풜 and consider the mappings ℱ and 풢 : 풜 → 풬 m ⁢ s ⁢ (풜) satisfying the relations (m + n) ⁢ ℱ ⁢ (a 2) = 2 ⁢ m ⁢ ℱ ⁢ (a) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ (a) and (m + n) ⁢ 풢 ⁢ (a 2) = 2 ⁢ m ⁢ 풢 ⁢ (a) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ (a) for all a ∈ 풜 . Using the theory of functional identities and the structure of involutions on matrix algebras, we prove that if ℱ and 풢 are additive, then 풢 = 0 . We also show that, in case ' * ' is any nonidentity anti-automorphism, the same conclusion holds if either ' * ' is not identity on 풵 ⁢ (풜) or 풜 is a PI-ring. [ABSTRACT FROM AUTHOR]

In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that every square matrix over an infinite field is always representable as a sum of a diagonalizable matrix and a nilpotent matrix of order less than or equal to two. In addition, each 2 × 2 matrix over any field admits such a representation. We, moreover, show that, for all natural numbers n ≥ 3, every n × n matrix over a finite field having no less than n + 1 elements also admits such a decomposition. The latter completes a recent example due to Breaz [Matrices over finite fields as sums of periodic and nilpotent elements. Linear Algebra Appl. 2018;555:92–97]. As a consequence of these decompositions, we show that every nilpotent matrix over a field can be expressed as the sum of a potent matrix and a square-zero matrix. This somewhat improves on recent results due to Abyzov et al. [On some matrix analogues of the little Fermat theorem. Mat Zametki. 2017;101(2):163–168] and Shitov [The ring M 8 k + 4 (Z 2) is nil-clean of index four. Indag Math (N.S.). 2019;30:1077–1078]. [ABSTRACT FROM AUTHOR]

We introduce the h-minimum spanning length of a family A of n × n matrices over a field F, where h is a p-tuple of positive integers, each no more than n. For an algebraically closed field F, Burnside's theorem on irreducibility is essentially that the (n,n , . . . , n)-minimum spanning length of A exists if A is irreducible. We show that the h-minimum spanning length of A exists for every h = h1,h2, . . . , hp) if A is an irreducible family of invertible matrices with at least three elements. The (1,1, . . . ,1)-minimum spanning length is at most 4n log2 2n + 8n − 3. Several examples are given, including one giving a complete calculation of the (p,q)-minimum spanning length of the ordered pair (J*,J), where J is the Jordan matrix. [ABSTRACT FROM AUTHOR]