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For every $ n \in \mathbb {N} $ n ∈ N and every field K, let $ A(n,K) $ A (n , K) be the vector space of the antisymmetric $ (n \times n) $ (n × n) -matrices over K. We say that an affine subspace S of $ A(n,K) $ A (n , K) has constant rank r if every matrix of S has rank r. Define $$\begin{align*} {\mathcal{A}}_{antisym}^K(n;r)& = \{ S \;| \; S \; {\rm affine \; subspace \; of } \; A(n,K) \; {\rm of \; constant \; rank } \; r \}, \\ a_{antisym}^K(n;r) & = \max \{\dim S \mid S \in {\mathcal{A}}_{antisym}^K(n;r) \}. \end{align*}$$ A antisym K (n ; r) = { S | S affine subspace of A (n , K) of constant rank r } , a antisym K (n ; r) = max { dim ⁡ S ∣ S ∈ A antisym K (n ; r) }. In this paper, we prove the following formulas: for $ n \geq ~2r +2 $ n ≥ 2 r + 2 \[ a_{antisym}^{\mathbb{R}}(n; 2r) = (n-r-1) r ; \] a antisym R (n ; 2 r) = (n − r − 1) r ; for n = 2r \[ a_{antisym}^{\mathbb{R}}(n; 2r) =r(r-1) ; \] a antisym R (n ; 2 r) = r (r − 1) ; for n = 2r + 1 \[ a_{antisym}^{\mathbb{R}}(n; 2r) = r(r+1). \] a antisym R (n ; 2 r) = r (r + 1). [ABSTRACT FROM AUTHOR]

We propose an effective matrix iteration method and a novel zeroing neural network (NZNN) model for finding numerical commuting solutions of the (time-invariant) Yang–Baxter-like matrix equation in this paper. The proposed matrix iteration method has a second-order convergence speed, and it is proved to be stable. How to proceed with initial value selection and termination criteria are discussed. Meanwhile, two numerical experiments are adopted to illustrate the superiority of the proposed matrix iteration method in computational efficiency. The NZNN model based on Tikhonov regularization for solving the time-invariant Yang–Baxter-like matrix equation is given. Besides, numerical results are provided to substantiate the efficiency, availability and superiority of the developed NZNN model for time-invariant Yang–Baxter-like matrix equation problems. [ABSTRACT FROM AUTHOR]

It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of attitude-orbit tracking control for flexible spacecraft. A relative attitude-orbit-structure integrated dynamics model is derived for flexible spacecraft, where environmental disturbances and parameter uncertainty are considered as lumped disturbance. And the relative position and attitude of the spacecraft are described by the exponential coordinates of SE(3). Since the modal variables are not measurable, a modal observer is proposed to obtain the state estimation related to elastic vibration. Then, a composite control technique is proposed for attitude-orbit tracking of flexible spacecraft under lumped disturbance by combining a state observer of modal parameter and nonlinear disturbance observer with an asymptotic tracking control. The vibration mode information and the lumped disturbance are estimated and compensated by the modal observer and the nonlinear disturbance observer, respectively, in the feedback link. The stability of the composed control approach consisting of the asymptotic tracking control and multi-observer observer is guaranteed through Lyapunov method. The simulation results validate the composite control technique can effectively enhance disturbance attenuation ability, robust dynamics performance and the desired relative attitude tracking accuracy of a flexible spacecraft with multiple disturbances and parameter uncertainty. [ABSTRACT FROM AUTHOR]

In this paper, we discuss centralizers in the Riordan group. We will see that Faà di Bruno's formula is an application of the Fundamental Theorem of Riordan arrays. Then the composition group of formal power series in F 1 is studied to construct the centralizers of Bell type and Lagrange type Riordan arrays. Our tools are the A-sequences of Riordan arrays and Faà di Bruno's formula. Some combinatorial explanation and discussion about related algebraic topics are also given. [ABSTRACT FROM AUTHOR]

Following the popularity of the core-EP (c-EP) and weighted core-EP (w-c-EP) inverses, so called one-sided versions of the w-c-EP inverse are introduced recently in Behera et al. (Results Math 75:174 (2020). These extensions are termed as E-w-c-EP and F-w-d-c-EP g-inverses as well as the star E-w-c-EP and the F-w-d-c-EP star classes of g-inverses. The applicability of these g-inverses in solving certain restricted matrix equations has been verified. Several additional results on these classes of g-inverses are established in this paper. In addition, the Moore–Penrose E-w-c-EP inverse and the F-w-d-c-EP Moore–Penrose inverse are proposed using proper expressions that involve the Moore–Penrose inverse and the E-w-c-EP or F-we-d-c-EP inverse. Further, the W-weighted Moore–Penrose c-EP and the W-weighted c-EP Moore–Penrose g-inverses are considered with the aim to extend the considered w-c-EP generalized inverses to rectangular matrices. Characterizations, properties, representations and applications of these inverses are considered. [ABSTRACT FROM AUTHOR]

Let R be non-commutative ring with unity and n ≥ p ≥ 2 , p prime. In this paper, we prove that an n × n matrix over R is the sum of pth powers if and only if its trace can be written as a sum of pth powers and commutators modulo pR. This extends the results of L. N. Vaserstein (p = 2) and S. A. Katre, Kshipra Wadikar (p = 3). We also obtain necessary and sufficient conditions for a matrix over R to be written as a sum of fourth powers when n ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a description of the derivatives in U T M n (S) , the semiring of upper triangular matrices over an additively idempotent semiring S. The main result states that an arbitrary derivation in the semiring U T M n (S) is a linear combination of finite number of well-known derivations. [ABSTRACT FROM AUTHOR]

In this paper, we give refinements for the G-majorization triangle inequality (x + y) ↓ ≺ G x ↓ + y ↓ with vectors x , y of an Eaton triple and normal map (⋅) ↓ . For instance, we establish refinements for the majorization Ky Fan's eigenvalue inequality λ (x + y) ≺ m λ (x) + λ (y) with Hermitian matrices x and y and the eigenvalue operator λ (⋅). Likewise, we show refinements for the weak majorization Ky Fan's singular value inequality s (x + y) ≺ w s (x) + s (y) with complex matrices x and y, where s (⋅) stands for the singular values operator. In particular, we recover original Maligranda's refinement of the norm triangle inequality. [ABSTRACT FROM AUTHOR]

S. A. Katre and Anuradha Garge obtained necessary and sufficient trace conditions for matrices over commutative rings with unity to be sums of kth powers. In this paper, we show that these conditions hold for matrices over noncommutative rings too. For n ≥ 2 , we deduce Vaserstein's result for sums of squares of matrices and also obtain nice trace conditions for n × n matrices to be sums of cubes. For an integer n ≥ k ≥ 2 , we get a sufficient condition (m o d k) for an n × n matrix over a noncommutative ring to be a sum of kth powers. [ABSTRACT FROM AUTHOR]

The aim of the paper is to present a procedure for the approximation of a symmetric positive definite matrix by symmetric block partitioned matrices with structured off-diagonal blocks. The entropy loss function is chosen as approximation criterion. This procedure is applied in a simulation study of the statistical problem of covariance structure identification. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notation of E-weighted core-EP and F-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses. Further, we discuss the existence of generalized weighted Moore–Penrose inverse and additive properties of the weighted core-EP inverse of matrices. In addition to these, we propose the star weighted core-EP and weighted core-EP star class of matrices for solving the system of matrix equations. We further elaborate on this theory by producing a few representation and characterization of star weighted core-EP and weighted core-EP star classes of matrices. [ABSTRACT FROM AUTHOR]

This paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems. [ABSTRACT FROM PUBLISHER]

Letandbe two Hilbert spaces, and letbe the algebra of all bounded linear operators frominto. We say that an elementis an all-derivable point inif every derivable linear mappingat(i.e.for anywith) is a derivation. Let bothandbe two linear mappings. In this paper, the following result will be proved : iffor anyand, thenandfor some. As an important application, we will show that an operatoris an all-derivable point inif and only if. [ABSTRACT FROM AUTHOR]

Letbe a finite field containingelements, and letbe the set ofstrictly upper triangular matrices over. In this paper, nonlinear bijective maps onpreserving zero product in both directions are explicitly described. [ABSTRACT FROM PUBLISHER]

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. In this paper, we present a new technique to compare the energies of two bipartite graphs whose characteristic polynomials satisfy a given recurrence relation. As its applications, we can characterize the bipartite unicyclic graphs of order $$n$$ with maximal, second-maximal, and third-maximal energy for $$n\ge 27$$ , and thus confirm the validity of the results in (Gutman et al. MATCH Commun Math Comput Chem 58:75-82, ) which were obtained by numerical calculations. [ABSTRACT FROM AUTHOR]

Continuous-time Markov chains are a standard tool in phylogenetic inference. If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain. In applications, there are usually extra constraints on the rates, depending on the situation. If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints. It is then useful to require that, under some time restrictions, there exists a homogeneous average of this inhomogeneous process within the same model. This leads to the definition of 'Lie Markov models' which, as we will show, are precisely the class of models where such an average exists. These models form Lie algebras and hence concepts from Lie group theory are central to their derivation. In this paper, we concentrate on applications to phylogenetics and nucleotide evolution, and derive the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines-that is, models with purine/pyrimidine symmetry. We also discuss how to handle the subtleties of applying Lie group methods, most naturally defined over the complex field, to the stochastic case of a Markov process, where parameter values are restricted to be real and positive. In particular, we explore the geometric embedding of the cone of stochastic rate matrices within the ambient space of the associated complex Lie algebra. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of infinite B*-matrices and linear operators over normed fields. Their structure is studied in the paper. Ideals and centers of the corresponding to them B*-algebras are scrutinized. [ABSTRACT FROM AUTHOR]

In this paper, we give a new angle to interpret Riordan arrays by showing that every Riordan array can be expressed as a Krylov matrix. We then use this idea to obtain some groups containing the Riordan group as a subgroup. Moreover, we study Lie algebras for the extended Riordan groups as Lie groups. [ABSTRACT FROM AUTHOR]

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [3] , [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals any power of 2, then it is similar to a permutation matrix group. [ABSTRACT FROM AUTHOR]

The lengths of matrix incidence algebras are studied when their radicals have square zero. All realizable values of the length function are provided for such algebras. In order to obtain this result, a discrete optimization problem is posed and solved. Also, the exact formula of the length is deduced under the additional assumption that the algebra is maximal by inclusion. Moreover, the solution to the length realizability problem is established for matrix incidence algebras with arbitrary radicals under a restriction on the cardinality of the ground field. [ABSTRACT FROM AUTHOR]

Let A be a d × d matrix with rational entries which has no eigenvalue λ ∈ C of absolute value | λ | < 1 and let Z d [ A ] be the smallest nontrivial A-invariant Z -module. We lay down a theoretical framework for the construction of digit systems (A , D) , where D ⊂ Z d [ A ] finite, that admit finite expansions of the form x = d 0 + A d 1 + ⋯ + A ℓ − 1 d ℓ − 1 (ℓ ∈ N , d 0 , ... , d ℓ − 1 ∈ D) for every element x ∈ Z d [ A ]. We put special emphasis on the explicit computation of small digit sets D that admit this property for a given matrix A, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way. [ABSTRACT FROM AUTHOR]

Let A and B be unital rings. An additive map T : A → B is called a weighted Jordan homomorphism if c = T (1) is an invertible central element and c T (x 2) = T (x) 2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = M n (R) with n ≥ 2 and R any unital ring with 1 2 , under which every surjective additive map T : A → B with the property that T (x) T (y) + T (y) T (x) = 0 whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char (A) ≠ 2 , 3 , 5 , then a bijective additive map T : A → A is a weighted Jordan homomorphism provided that there exists an additive map S : A → A such that S (x 2) = T (x) 2 for all x ∈ A. [ABSTRACT FROM AUTHOR]

Using a standard embedding of K[x], the algebra of polynomials with coefficients in a field K, into the algebra of row and column finite matrices over K, non-trivial factorizations of some irreducible polynomials in K[x] are possible in the larger context. The row and column finite matrices involved in those factorizations resemble polynomials in several ways. In fact, each one of them is induced by a finite number of polynomials. We call these matrices entangled polynomials. For every fixed m ∈ Z + , entangled polynomials induced by m polynomials (the so-called m-nomials) form a ring, K m [ x ] . The ring of 1-nomials is precisely K[x]. When n divides m, K n [ x ] is a subring of K m [ x ] . In particular, all rings of m-nomials include the polynomials. We consider alternative representations of entangled polynomials as finite square matrices. These representations allow us to use determinants to characterize units and irreducible elements in the various rings of entangled polynomials. A cancellation monoid, the homogeneous m-nomials, and its properties are also considered. [ABSTRACT FROM AUTHOR]

We study smooth quadric surfaces in the Pfaffian hypersurface in P 14 parameterizing 6 × 6 skew-symmetric matrices of rank at most 4, not intersecting its singular locus. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle E on the quadric. We analyse these bundles and their geometry, relating them to linear congruences in P 5 . [ABSTRACT FROM AUTHOR]

In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems. [ABSTRACT FROM AUTHOR]

Let M n ( F ) be the algebra of n × n matrices 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. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed. [ABSTRACT FROM AUTHOR]

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max ⁡ n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. Refs. [3] and [4] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a prime or a square of a prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle of length equal to any power of any odd prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. [ABSTRACT FROM AUTHOR]

Every transitive family of subspaces of a vector space of finite dimension $n\ge 2$ over a field $\mathbb {F}$ contains a subfamily which is transitive but has no proper transitive subfamily. Such a subfamily is called minimally transitive. Each has at most $n^2-n+1$ elements. On ${{\mathbb {C}}}^n, n\ge 3$ , a minimally transitive family of subspaces has at least four elements and a minimally transitive family of one-dimensional subspaces has $\tau $ elements where $n+1\le \tau \le 2n-2$. We show how a minimally transitive family of one-dimensional subspaces arises when it consists of the subspaces spanned by the standard basis vectors together with those spanned by $0$ – $1$ vectors. On a space of dimension four, the set of nontrivial elements of a medial subspace lattice has five elements if it is minimally transitive. On spaces of dimension $12$ or more, the set of nontrivial elements of a medial subspace lattice can have six or more elements and be minimally transitive. [ABSTRACT FROM AUTHOR]

We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad consists of three diagonalizable linear transformations on a finite-dimensional vector space which satisfy the following condition. The eigenspaces of the three transformations can be ordered so that each transformation raises the eigenspaces of the other two in a block-bidiagonal fashion. We show that each of the three eigenvalue sequences associated to a bidiagonal triad satisfy the same linear recurrence relation. Based on the solutions to this recurrence we define the notion of a reduced bidiagonal triad, and show that every bidiagonal triad is equivalent to a reduced one. We show that every finite-dimensional irreducible representation of the tetrahedron Lie algebra provides numerous examples of reduced bidiagonal triads. Conversely, we show how irreducible representations of the tetrahedron algebra can be constructed starting from a certain type of reduced bidiagonal triad. [ABSTRACT FROM AUTHOR]

In factoring matrices into the product of two matrices, operations are typically performed with elements restricted to matrix subspaces. Such modest structural assumptions are realistic, for example, in large scale computations. This paper is concerned with analyzing associated matrix geometries. Curvature of the product of two matrix subspaces is assessed. The case of vanishing curvature gives rise to the notion of factorizable matrix subspace. This can be regarded as an analogue of the internal Zappa–Szép product in group theory. Interpreted in this way, several classical instances are encompassed by this structure. The Craig–Sakamoto theorem fits naturally into this framework. [ABSTRACT FROM AUTHOR]

In this paper, we discuss tripotent1 elements in the finite ring ℍ/ℤp. We provide examples and establish conditions for tripotency. We follow similar methods used in [3] for idempotent elements in ℍ/ℤp.

Frobenius' Theorem states that, besides the fields of real and complex numbers, the algebra of quaternions H is the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then characterize finite-dimensional real algebras that contain either a copy of C , a copy of H , or a pair of anticommuting invertible elements through the dimensions of their (left) ideals, and finally consider the problem of lifting algebraic elements modulo ideals. [ABSTRACT FROM AUTHOR]

The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra A . After obtaining some general results, we classify all the maximal algebras for certain particular cases of A . [ABSTRACT FROM AUTHOR]

The semi-tensor product, which was initially proposed by Cheng et al. [An introduction to semi-tensor product of matrices and its applications. World Scientific; 2012], has been extensively applied in Boolean control networks, graph colouring, game theory, cryptographic algorithms and so on. In this article, motivated by the existing work by Yao et al. [J Franklin Inst. 2016;353:1109–1131], we further investigate the solvability of the matrix equation AXB=C with respect to semi-tensor product. The case of matrix-vector equation, in which the required unknown X be a vector, is studied first. Compatible condition for matrix dimensions, necessary and sufficient conditions and concrete solving methods are established. Based on this, the solvability of the matrix equation case, in which the unknown X be a matrix, under semi-tensor product is then studied. For each part, several elementary examples are presented to illustrate the efficiency of the results. [ABSTRACT FROM AUTHOR]

The connection of different modifications of the method of linear integral representation is studied. Solutions of the related inverse problems based upon a 'hybrid version of two approximations' of the topography and geopotential fields enable more refined tuning of the method in solving the inverse problems of geophysics and geomorphology and more complete allowance for the a priori information about the surface elevation data and elements of anomalous fields. The technique for finding a stable approximate solution for the inverse problem of determining the mass distributions equivalent by the external gravitational (or any other potential) field is presented. The method is applied to study anomalous densities beneath one of the highest elevation Elysium Mons and deep Hellas basin on Mars. The results of the mathematical experiment are discussed. [ABSTRACT FROM AUTHOR]

It is known that the partial entanglement/separability violates distributive rules with respect to the operations of taking convex hull and intersection. In this note, we give criteria for three-qubit partially entangled states arising from distributive rules, together with the corresponding witnesses. The criteria will be given in terms of diagonal and anti-diagonal entries. They actually characterize those partial entanglement completely when all the entries are zero except for diagonal and anti-diagonal entries. Important states like Greenberger–Horne–Zeilinger diagonal states fall down in this class. [ABSTRACT FROM AUTHOR]

We show that an irreducible family ${\mathcal{S}}$ of complex $n\times n$ matrices satisfies Paz's conjecture if it contains a rank-one matrix. We next investigate properties of families of rank-one matrices. If ${\mathcal{R}}$ is a linearly independent, irreducible family of rank-one matrices then (i) ${\mathcal{R}}$ has length at most $n$ , (ii) if all pairwise products are nonzero, ${\mathcal{R}}$ has length 1 or 2, (iii) if ${\mathcal{R}}$ consists of elementary matrices, its minimum spanning length $M$ is the smallest integer $M$ such that every elementary matrix belongs to the set of words in ${\mathcal{R}}$ of length at most $M$. Finally, for any integer $k$ dividing $n-1$ , there is an irreducible family of elementary matrices with length $k+1$. [ABSTRACT FROM AUTHOR]

We found three qubit Greenberger–Horne–Zeilinger diagonal states which tell us that the partial separability of three qubit states violates the distributive rules with respect to the two operations of convex sum and intersection. The gaps between the convex sets involving the distributive rules are of nonzero volume. [ABSTRACT FROM AUTHOR]

Our aim is to establish relations between the Drazin inverses of the pesudo-block matrix (P , Q , R , S) and the block matrix P R S Q , where R 2 = S 2 = 0. Based on the relations, we give representations for the Drazin inverse of the sum P + Q + R + S under mild restrictions. As the applications, several expressions for the Drazin inverse of a 2 × 2 block matrix are presented under some assumptions. Our results generalize several results in the literature. [ABSTRACT FROM AUTHOR]

Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$ , where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$. [ABSTRACT FROM AUTHOR]

Motivated by comatrix coalgebras, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on matrix algebras, via the construction of a suitable coproduct. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on matrix algebras. By the close relationship between pre-Lie algebras and infinitesimal unitary bialgebras, we erect a pre-Lie algebra and a new Lie algebra on matrix algebras. Finally, a weighted infinitesimal unitary bialgebra on non-commutative polynomial algebras is also given. [ABSTRACT FROM AUTHOR]

A classification of full algebras of matrices is given. All such algebras are permutation-isomorphic to block lower-triangular matrices with corresponding subdiagonal blocks being either zero-blocks or full. Two full algebras are isomorphic if and only if they are permutation-isomorphic. A one-to-one correspondence is provided between the full algebras and transitive directed graphs. It is also proven that such algebras, if endowed with a lattice order, can be almost f- or d-algebras only if they are diagonal. [ABSTRACT FROM AUTHOR]