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A square matrix A has the usual Jordan canonical form that describes the structure of A via eigenvalues and the corresponding Jordan blocks. If A is a linear relation in a finite-dimensional linear space \mathfrak {H} (i.e., A is a linear subspace of \mathfrak {H} \times \mathfrak {H} and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product \mathfrak {H} \times \mathfrak {H}), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation A. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue \infty; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation. [ABSTRACT FROM AUTHOR]

We study the categories of matrix representations of nilpotent semigroups over an arbitrary field. Our main attention is given to the Auslander matrix algebras as one of possible forms of specifying the categories of representations, which have finitely many equivalence classes of indecomposable objects, and to the analysis of their discrete characteristics. The concept of Auslander algebra is generalized to the category of representations without auxiliary conditions. For any nilpotent cyclic semigroup, we describe its Auslander algebra and determine its Σ -function. [ABSTRACT FROM AUTHOR]

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]

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ (A) + δ (A) A − 1 + A τ (A − 1) + τ (A − 1) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ (A) = D (A) + A δ (I) and τ (A) = D (A) − δ (I) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ (A) B + A τ (B) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M. Similar conclusions are obtained for additive mappings from a matrix ring M n (D) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 . [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]

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 each integer n ≥ 3 we define a specific symmetric integer matrix of size φ (n) / 2 . We then explicitly describe all positive and negative integer powers, as well as fractional powers, of each of these matrices; their eigenvalues also appear and play an important role. All this is based on an orthogonality relation for odd Dirichlet characters which, in turn, leads to a diagonalization. In the course of this article, various other number-theoretic objects make their appearance, including the Euler and Jordan totient functions, Möbius inversion, and the divisor function. [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 problem of extensions of the canonical Lee–Poisson–Kirillov–Kostant symplectic structure of coadjoint orbit of the complex general linear group is considered. The introduced method uses the concept of flag coordinates and does not depend on the Jordan structure of the matrices that form the orbit. The principal bundle associated with the fibration of the orbit over the Grassmanian of flags is constructed. [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]

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]

If the cosquares of complex matrices and are similar and there is a unimodular number in the spectrum of the cosquares, then and are not necessarily congruent. Assume that such an eigenvalue is unique. In this case, so far, one could verify the congruence of and by using a rational algorithm only in two situations: (1) the eigenvalue is simple or semi-simple; (2)there is only one Jordan block associated with in the Jordan form of the cosquares. We propose a rational algorithm for checking congruence in the case where two Jordan blocks of the same order are associated with . [ABSTRACT FROM AUTHOR]

In this paper, we first define a new regular matrix by using the arithmetic function called Jordan totient function and study the matrix domain of this newly introduced matrix in the Banach space ℓp. After computing the dual spaces of this new space, we characterize certain matrix mappings related to this space. [ABSTRACT FROM AUTHOR]

In this paper, we first define a new regular matrix by using the arithmetic function called Jordan totient function and study the matrix domain of this newly introduced matrix in the Banach space ℓp. After computing the dual spaces of this new space, we characterize certain matrix mappings related to this space. [ABSTRACT FROM AUTHOR]

In the present study, we introduce a new RH-regular 4D (4-dimensional) matrix derived by Jordan's function and define double sequence spaces by using domains of 4D Jordan totient matrix J t on some classical double sequence spaces. Also, the α -, β ϑ -, and γ -duals of these spaces are determined. Finally, some classes of 4D matrices on these spaces are characterized. [ABSTRACT FROM AUTHOR]

In our prior studies, we synthesized special circuits with evolution matrices featuring degenerate eigenfrequencies and nontrivial Jordan blocks. The degeneracy of this type is sometimes referred to as exceptional point of degeneracy (EPD). Our focus here is on the simplest of our circuits featuring EPDs that are composed of only two LC-loops coupled by a gyrator. These circuits, when near an EPD state, can be used for enhanced sensitivity applications. With that in mind, we develop here a comprehensive perturbation theory for these simple circuits near an EPD. Using this theory, we propose an approach to sensing, allowing one to benefit from the proximity to an EPD on the one hand when providing for stable operation on the other hand. We also address a broader scope of problems related to perturbations of Jordan blocks and their numerical treatment that allow us to effectively detect proximity to Jordan blocks. [ABSTRACT FROM AUTHOR]

We analyze the solutions of one-sided matrix equations over algebraically closed fields of characteristic zero in a certain transcendental extension of the main field that are generated by a set of invertible matrices. We establish conditions under which the roots corresponding to different invertible matrices from this set coincide. [ABSTRACT FROM AUTHOR]

For a class of second-order cone nonlinear complementarity problems, abbreviated as SOCNCPs, we establish the modulus-based matrix splitting relaxation iteration methods, which are obtained by reformulating equivalently SOCNCP as an implicit fixed-point equation based on Jordan algebra associated with the second-order cone. The global convergence theorems are given under suitable choices of the involved splitting matrix and parameter matrices. When the splitting matrix is symmetric positive definite, the strategy choice of the parameters is discussed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SOCNCPs. [ABSTRACT FROM AUTHOR]

In this paper, we study the noncommutative functional equation h (λ z) - λ h (z) = g (z) , z ∈ C and we give a new perspective from this equation to obtain a completely dynamical classification for complex matrices. Coarsely speaking, there are four different types: 0 , 1 2 , 2 and e i 2 π θ with θ ∈ [ 0 , 1 2 ] for diagonal matrices and Jordan matrices, respectively. Moreover, we obtain that for a complex matrix A, if its eigenvalues are 0 < | λ i | ≠ 1 , where 1 ≤ i ≤ rank (A) , then A is topologically conjugate to a diagonal matrix on C rank (A) . [ABSTRACT FROM AUTHOR]

We study the Schwarz problem for J-analytic vector-valued functions in an ellipse with a square matrix J admitting a nondiagonal Jordan form. We obtain conditions on the ellipse and matrix J necessary and sufficient for the existence and uniqueness of a solution to the Schwarz problem with an arbitrary boundary function of Hölder class. Under certain conditions on the matrix J, we show that the homogeneous Schwarz problem in an ellipse has a solution in the form of a vector polynomial of an arbitrary degree. [ABSTRACT FROM AUTHOR]

We advance here an algorithm of the synthesis of lossless electric circuits such that their evolution matrices have the prescribed Jordan canonical forms subject to natural constraints. Every synthesized circuit consists of a chain-like sequence of LC-loops coupled by gyrators. All involved capacitances, inductances, and gyrator resistances are either positive or negative with values determined by explicit formulas. A circuit must have at least one negative capacitance or inductance for having a nontrivial Jordan block for the relevant matrix. [ABSTRACT FROM AUTHOR]

The problem of finding the distance from a given n×n matrix polynomial of degree k to the set of matrix polynomials having the elementary divisor (λ-λ0)j,j⩾r, for a fixed scalar λ0 and 2⩽r⩽kn is considered. It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor. For regular matrix polynomials the problem is shown to be equivalent to finding minimal structure preserving perturbations such that a certain block Toeplitz matrix becomes suitably rank deficient. This is then used to characterize the distance via two different optimizations. The first one shows that if λ0 is not already an eigenvalue of the matrix polynomial, then the problem is equivalent to computing a generalized notion of a structured singular value. The distance is computed via algorithms like BFGS and Matlab's globalsearch algorithm from the second optimization. Upper and lower bounds of the distance are also derived and numerical experiments are performed to compare them with the computed values of the distance. [ABSTRACT FROM AUTHOR]

We study the Schwarz problem for J-analytic functions in the case where the Jordan basis Q of the matrix J has real vectors and the Jordan form J1 of J contains Jordan (2×2)-cells, whereas the order of real columns of the matrix Q depends on the structure of J1. We prove the existence and uniqueness of a solution to the Schwarz problem in Hölder classes. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & Astr. (2019) and Chebyshevskii Sb. (2019), respectively. [ABSTRACT FROM AUTHOR]

A general method is given for merging blocks in the Jordan canonical form of a nonnegative matrix. As a consequence, results, more general than any prior ones, are given for the universal realizability of spectra, that is, spectra which are realizable by a nonnegative matrix for each possible Jordan canonical form allowed by the spectrum. In particular, we generalize a classical result due to Minc, regarding positive diagonalizable matrices. For example, any spectrum that is diagonalizably realizable by a nonnegative matrix with mostly positive off-diagonal entries is universally realizable. [ABSTRACT FROM AUTHOR]

Differentiation matrices are in wide use in numerical algorithms, although usually studied in an ad hoc manner. We collect here in this review paper elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange and Hermite interpolational bases. We give new explicit formulations, and new explicit formulations for the pseudo-inverses which help to understand antidifferentiation, of many of these matrices. We also give the unique Jordan form for these (nilpotent) matrices and a new unified formula for the transformation matrix. [ABSTRACT FROM AUTHOR]

Let S be an n-by-n, nonsingular, and Hermitian matrix. A square complex matrix Q is said to be S-unitary if Q*SQ = S. An S-unitary matrix Q is said to be elementary if rank(Q -- I) = 1. It is known what form every elementary S-unitary can take, and that every S-unitary can be written as a product of elementary S-unitaries. In this paper, we determine the Jordan canonical form of a product of two elementary S-unitaries. [ABSTRACT FROM AUTHOR]

We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A , B ∈ GL m (C) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k ⩾ 0 the matrix A B k is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈 A , B 〉 < GL m (C) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces. [ABSTRACT FROM AUTHOR]

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The problem of the similarity of integer matrices with single eigenvalue over the ring of integers is considered. A criterion for a matrix to be similar to a Jordan block is obtained. In addition, a similarity criterion for matrices whose minimal polynomial has degree 2 is obtained. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization. [Copyright &y& Elsevier]

Many efforts have been made for model predictive control MPC to enlarge terminal region where the stability is guaranteed by including a proper terminal weighting. This paper investigates how to maximize the ellipsoidal terminal region by using a more general method to design MPC: calculating terminal region and terminal weighting separately. With Jordan canonical form, the new ellipsoid-based method can result in terminal regions which are infinite (in certain directions) even in the case of open-loop unstable systems. Some most popular methods are just special cases of this new method. Simulation results illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for the computation of the Jordan canonical form of regular matrix polynomials is proposed. The new method contains rank conditions of suitably defined block Toeplitz matrices and it does not require the computation of the Jordan chains or the Smith form. The Segré and Weyr characteristics are also considered. [Copyright &y& Elsevier]

A brief review of the essentials behind the construction of a Chern-Simons-like brane action from the Large N limit of Exceptional Jordan Matrix Models paves the way to the construction of actions for membranes and p -branes moving in octonionic-spacetime backgrounds endowed with octonionic-valued metrics. It is found that the action of a membrane moving in spacetime backgrounds endowed with an octonionic-valued metric is not invariant under the usual diffeomorphisms of its world volume coordinates σ a → σ ′ a (σ b) , but instead, it is invariant under the rigid E 6 (− 26) transformations which preserve the volume (cubic) form. An extensive study of the bosonic octonionic string moving in flat backgrounds, and its quantization, follows. One of the most significant findings (without invoking supersymmetry) is that the number of degrees of freedom of three (massless) fermion generations is the same as the number of transverse dimensions of an octonionic string moving in D = 26 octonionic dimensions. The star-product deformations of octonionic p -branes follow. In particular, we focus on the octonionic membrane along with the phase space quantization methods developed by [30] within the context of Nonassociative Quantum Mechanics. We finalize with an analysis of two octonionic gravitational actions and with some concluding remarks on Double and Exceptional Field theories, Nonassociative Gravity and A ∞ , L ∞ algebras. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the eigenproblem on the large and low-rank matrix S = A B H , where A , B ∈ ℂ n × r are of full column rank and r ≪ n. To the best of our knowledge, there are no results on the relations between the Jordan decomposition and the Schur decomposition of B H A and those of A B H . Some known results are only on characteristic polynomials, elementary divisors, and Jordan blocks of A B H , and are purely theoretical and are not easy to use for computational purposes. Based on the Jordan decomposition and the Schur decomposition of the small matrix B H A ∈ ℂ r × r , we consider how to derive those of the large matrix A H B ∈ ℂ n × n in this work. The construction methods proposed are not only theoretical but also practical. Numerical experiments show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]