In this article, we study special $ 2 imes n $ 2 & times;n row Suslin matrices $ S_n<^>2(V,W) $ Sn2(V,W), arising from the pair of matrices $ V,W \in M_{2 ,n}(R) $ V,W is an element of M2,n(R) with $ V.W<^>T = I_2 $ V.WT=I2. These matrices generate a subgroup $ SUm_n<^>2(R) $ SUmn2(R) of the general linear group $ GL_{2<^>{2n-1}}(R) $ GL22n-1(R). Our primary focus is on the elementary subgroup $ EUm_n<^>2(R) \subset SL_{2<^>{2n-1}}(R) $ EUmn2(R)subset of SL22n-1(R) of $ SUm_n<^>2(R) $ SUmn2(R), for which we determine a complete set of generators by establishing the key lemma. We further investigate the algebraic properties of these generators and prove a structure theorem for $ EUm_n<^>2(R) $ EUmn2(R) that simplifies the generating set. The findings presented in this article enhance the understanding of matrix group generation and have potential implications for algebraic K-theory and related areas.
Assume that M is a Hilbert C*-module over a C* -algebra A . In this paper, we first propose the concepts of generalized derivations and generalized Jordan derivations on M , and then by giving some useful properties of M , we prove that C -linear generalized (Jordan) derivations on M are automatically continuous under the norm topology. In addition, we show that, for any C -linear map delta on M , if A is commutative, then the following statements are equivalent: (1) delta is a generalized Jordan derivation, (2) delta is a generalized derivation, (3) delta is a Jordan derivation, (4) delta is a derivation; if delta is A -linear, M is full and A is unital, then delta is a generalized derivation if and only if delta is a derivation. Finally, we also discuss the structure of other generalized forms of (Jordan) derivations.
We construct the sequence spaces h(Delta((2))(q)) = h(Delta q(2)) and bv(Delta((2))(q)) = bv ((2))(Delta q), where Delta((2))(q) is a second order q-difference operator defined by (Delta((2))(q)u)(d) = u(d) - (1 + q)u(d- 1) + qu(d-2) for all d is an element of Z(0)(+), where we presume that u(d) = 0 for d <= 0. We obtain some inclusion relations, Schauder bases, and important duals of the spaces h(Delta((2))(q)) and bv(Delta((2))(q)). We also determine necessary and sufficient condition for a matrix to map between the spaces h(Delta((2))(q)) /bv(Delta((2))(q)) and any arbitrary space mu. In the final section, we determine the point spectrum and the spectrum of the operator Delta((2))(q) over the space h.
In this paper we establish necessary and sufficient condition for a signed Cayley graph to be strongly regular and present several constructions of strongly regular signed Cayley graphs, which are proven to be net-regular. Furthermore, we calculate their adjacency eigenvalues and demonstrate that they have at most four distinct eigenvalues.
This article examines a way to define the notion of left (right) $ (b,c) $ (b,c)-polar elements in an associative ring R. Necessary and sufficient conditions of an element $ a\in \mathcal {R} $ a is an element of R to be left (right) $ (b,c) $ (b,c)-polar are investigated. We show that an element $ a\in \mathcal {R} $ a is an element of R is left (right) $ (b,c) $ (b,c)-polar if and only if a is strongly left (right) $ (b,c) $ (b,c)-invertible. Moreover, the one-sided $ (b,c) $ (b,c)-polarity is characterized in terms of left and right annihilators. As an application, we provide a characterization of the one sided core inverse. Further results and properties are obtained.
Superregular matrices, i.e. matrices where all square submatrices are non-singular, have a wide range of applications in communications. A block superregular matrix is a broader concept where all full block submatrices, with the appropriate size, are non-singular. In this work we propose a construction of block superregular matrices based on the Kronecker product of a superregular matrix with a non-singular matrix. Furthermore, we propose two constructions of superregular matrices via other smaller superregular matrices over smaller fields.
The extended core inverse was presented for square complex matrices as an extension of the core inverse based on the sum and difference of known generalized inverses. Unlike of existing generalizations of the core inverse, the extended core inverse is an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. The aim of this paper is to consider a generalization of the system for defining the extended core inverse based on a weight and to introduce a weighted extended core inverse for operators between two Hilbert spaces. Thus, we consider a new type of weighted generalized inverses. Properties, characterizations and expressions for the weighted extended core inverse are established. The dual version of the weighted extended core inverse is studied too and it presents a new extension of the Moore-Penrose inverse. We apply the weighted extended core inverse and its dual to solve certain systems of linear equations and minimization problems.
Although the numerical radius is not sub-multiplicative, we tickle this problem by showing a weak sub-multiplicativity behaviour. This will be discussed for the usual numerical radius, and the recently defined generalized numerical radius for any unitarily invariant norm, defined on an ideal in the $ C<^>*- $ C & lowast;-algebra of bounded linear operators on a complex Hilbert space.
We express the eigenvectors of the Laplacian matrix of a graph as a linear combination of the eigenvectors of the effective resistance matrix of that graph and vice versa. We find partial fraction expansions for the eigenvalues of either matrix in terms of the spectrum of the other matrix as well as the explicit characteristic polynomials and some sums of powers of eigenvalues. We also improve the bounds of the Interlacing Theorem for the eigenvalues of the Laplacian and effective resistance matrix of a same graph.
Let A and B be two positive bounded linear operators acting on two complex Hilbert spaces H and K, respectively. In this paper, we study the (A circle times B)-maximal numerical range W-max(A circle times B) (T circle times S) of the tensor product T circle times S for two bounded linear operators T and S on H and K , respectively. In the context of this work, we show under some hyponormality conditions, the following equality W-max(A circle times B) (T circle times S) =co (W-max(A)(T) & centerdot; W-max(B)(S)) holds, where W-max(A)(T), W-max(B)(S) and co(& centerdot;) denote respectively the A-maximal numerical range of T, the B-maximal numerical range of S and convex hull. Furthermore, we extend Fong's result to the class of operators defined on the semi-Hilbertian space.
We define the first matrix-weighted alternating zeta function of a digraph D, and give its determinant expression. We present a decomposition formula for the first matrix-weighted alternating zeta function of a group covering of D. Furthermore, we introduce the first matrix-weighted alternating L-function of D, and present a determinant expression for it. As a corollary, we present a decomposition formula for the first matrix-weighted alternating zeta function of a group covering of D by its first matrix-weighted alternating L-functions.
We show that every non-invertible square matrix over a division ring D can be expressed as the product of a unipotent matrix and a nilpotent matrix. As an application, we further show that every non-invertible matrix over D can be expressed as the product of at most two matrices that lie in the image of some polynomial in one variable with coefficients from the centre of D.
The power of orthogonality over the real and complex numbers lies in its use in computational and numerical methods. In this article, we discuss two different quasi-inner products over the finite fields, provide illustrative examples and develop additional results on how self-orthogonal vectors affect properties of these quasi-inner product spaces. We point out an error in a published paper and provide a correction. We examine relationships between a subspace W and its orthogonal complement. We then discuss different types of bases for subspaces and their implications. Finally, we compare and contrast the properties of quasi-inner product spaces depending on whether the transpose or the conjugate transpose is used to define the quasi-inner product space.
We extend the classical Perron-Frobenius theory to tensors that may have negative entries, thereby broadening the scope of spectral analysis beyond the nonnegative setting. Under certain sufficient conditions, we establish the existence of a Perron-Frobenius eigenpair for such tensors and characterize the corresponding spectral radius. As an application, we propose a novel concept termed Perron-Frobenius splitting for tensors, which facilitates the solution of multi-linear systems via tensor splitting iterative methods. This framework generalizes the regular and weak regular splittings of the tensors. Furthermore, we provide convergence analyses and comparison theorems for the Perron-Frobenius splittings of the tensors.
