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A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of such a space when the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer what is perhaps the most general problem of the kind as proposed by Loewy and Radwan, by solving their conjecture in the positive. We give the maximal dimension of a vector space of $ n\times n $ n × n matrices with no more than k

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by M 4 × 4 = (M 4 × 4 − D M 4 × 4) ∪ D M 4 × 4 , where D M 4 × 4 is the set of all dependent metric matrices. D M 4 × 4 is further characterized by D M 4 × 4 = D M 1 4 × 4 ∪ D M 2 4 × 4 , where D M 2 4 × 4 is characterized by D M 2 4 × 4 = D M 21 4 × 4 ∪ D M 22 4 × 4 . These characterizations provide some insightful findings that go beyond the Euclidean distance or Euclidean distance matrix and link the distance functions to vector spaces, which offers some theoretical and application-related advantages. In the application parts, we show that the metric matrices associated with all Minkowski distance functions over four different points are linearly independent, and that the metric matrices associated with any four concyclic points are also linearly independent. [ABSTRACT FROM AUTHOR]

In recent years, research on attribute-based encryption (ABE) has expanded into the quantum domain. Because a traditional single authority can cause the potential single point of failure, an improved lattice-based quantum-resistant identity authentication and policy attribute encryption scheme is proposed, in which the generation of random values is optimized by adjusting parameters in the Gaussian sampling algorithm to improve overall performance. Additionally, in the key generation phase, attributes are processed according to their shared nature, which reduces the computational overhead of the authorization authority. In the decryption phase, the basis transformation of the Lenstra–Lenstra–Lovász (LLL) lattice reduction algorithm is utilized to rapidly convert shared matrices into the shortest vector form, which can reduce the computational cost of linear space checks. The experimental results demonstrate that the proposed method not only improves efficiency but also enhances security compared with related schemes. [ABSTRACT FROM AUTHOR]

We continue the study of truncated matrix-valued moment problems begun in [12]. Let q ∈ N. Suppose that (X , X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into H q , the Hermitian q × q matrices. A linear functional Λ on E is called a moment functional if there exists a positive H q -valued measure μ on (X , X) such that Λ (F) = ∫ X 〈 F , d μ 〉 for F ∈ E. In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from [11] to obtain a matricial version of the flat extension theorem. Assuming that X is a compact space and all elements of E are continuous on X we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem. [ABSTRACT FROM AUTHOR]

We study a matrix analog of the Erdős-Falconer distance problems in vector spaces over finite fields. We provide lower bounds for the cardinality of a subset of the matrix algebra over a finite field such that its distance set is the whole matrix algebra. There arises an interesting analysis of certain quadratic matrix Gauss sums. [ABSTRACT FROM AUTHOR]

In this article, the matrix algebra is well-known concept in mathematics, has been extended to sedenionic-coefficient matrices using sedenions, which have many applications in recent years. Subsequently, sedenionic-coefficient matrices have been obtained in their real, complex, quaternionic and octonionic forms. Based on these definitions, the arithmetic operations of addition, multiplication, conjugation, transpose and conjugate transpose for sedenionic matrices and their complex, quaternionic and octonionic variations have been established and their algebraic properties scrutinized. Lastly, vector space over real and complex numbers and module structure over quaternions of sedenionic matrices has been searched. [ABSTRACT FROM AUTHOR]

We introduce the α−higher rank form of the matrix numerical range, which is a special case of the matrix polynomial version of the higher rank numerical range. We also, investigate some algebraic and geometrical properties of this set for general and nilpotent matrices. Some examples to confirm the results are brought. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study sequence spaces of Cesàro–Nörlund operators of order n associated with a sequence of Orlicz functions. We obtain some topological properties and Schauder basis of these sequence spaces. Moreover, we compute the α-, β- and γ-duals and the matrix transformations of these newly formed sequence spaces. Finally, we prove that these sequence spaces are of Banach–Saks type p and have a weak fixed-point property. [ABSTRACT FROM AUTHOR]

The main goal of this research paper is to find an algebraic ring homomorphism between symbolic 2-plithogenic ring and the corresponding 2-cyclic refined ring. This work presents some applications of the defined homomorphism to explain some algebraic relationships between symbolic 2-plithogenic algebraic structures and 2-cyclic refined structures. [ABSTRACT FROM AUTHOR]

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤ k as well as k-positivity and k-block-positivity. We also compare de Pillis' definition [Pac. J. Math. 23, 129–137 (1967)] and Choi's definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating v and w , then there exists f ∈ S separating v and w. In this article we consider the action of G = SL 2 (C) × SL 2 (C) on the C -vector space M 2 , 2 n of n -tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S n of C [ M 2 , 2 n ] G are known, and | S n | = 1 24 (n 4 − 6 n 3 + 23 n 2 + 6 n). In recent work, Domokos [8] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 , 2 n ] G. Our main result shows that any separating set for C [ M 2 , 2 n ] G has cardinality ≥ 5 n − 9. In particular, there is no separating set of size dim ⁡ (C [ M 2 n ] G) = 4 n − 6 for n ≥ 4. Further, S 4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S n. We also consider the action of G = SL l (C) on M l , n by left multiplication. In that case the algebra of invariants has a minimum generating set of size ( n l ) (the l × l minors of a generic matrix) and dimension l n − l 2 + 1. We show that a separating set for C [ M l , n ] G must have size at least (2 l − 2) n − 2 (l 2 − l). In particular, C [ M l , n ] G does not contain a separating set of size dim ⁡ (C [ M l , n ] G) for l ≥ 3 and n ≥ l + 2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem. [ABSTRACT FROM AUTHOR]

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Given an algebra F[H]G of polynomial invariants of an action of the group G over the vector space H, a subset S of F[H]G is called separating if S separates all orbits that can be separated by F[H]G. A minimal separating set is found for some algebras of matrix invariants of several matrices over an infinite field of arbitrary characteristic different from two in case of the orthogonal group. Namely, we consider the following cases: • GL(3) -invariants of two matrices; • O(3) -invariants of d > 0 skew-symmetric matrices; • O(4) -invariants of two skew-symmetric matrices; • O(3) -invariants of two symmetric matrices. A minimal generating set is also given for the algebra of orthogonal invariants of three 3×3 symmetric matrices. [ABSTRACT FROM AUTHOR]

The distributed-order fractional telegraph models are commonly used to describe the phenomenas of diffusion and wave-like anomalous, which can model processes without a power-law scale across the entire temporal domain. To increase the range of implementation of distributed-order fractional telegraph models, there is a need to present effective and accurate numerical algorithms to solve these models, as these models are hard to solve analytically. In this work, a novel matrix representation of the distributed-order fractional derivative based on shifted Gegenbauer (SG) polynomials has been derived. Also, two efficient algorithms based on the aforementioned operatonal matrix and the spectral tau method have been constructed for solving the one- and two-dimensional (1D and 2D) distributed-order time-fractional telegraph models with spatial variable coefficients. Also, a new operational matrix of the multiplication of space vectors has been built to have the ability in applying the tau method in the 2D case. The convergence and error bound analysis of the presented techniques are investigated. Moreover, the presented algorithms are applied on four miscellaneous test examples to illustrate the robustness and effectiveness of these algorithms. [ABSTRACT FROM AUTHOR]

Let 1 ≤ p, q < ∞ be fixed, and let R = [ rjk ] be an infinite scalar matrix such that 1 ≤ rjk < ∞ and sup j,k rjk < ∞ . Let B(lp, lq) be the set of all bounded linear operator from lp into lq . For a fixed Banach algebra B with identity, we define a new vector space SR p,q (B) of infinite matrices over B and a paranorm G on SR p,q (B) as follows: let SR p,q (B) { A : A [R] ∈ B(lp, lq) } and G(A) = ∥ A [R] ∥ 1/M p,q, where A [R] = [∥ ajk ∥ rjk and M = max{ 1, supj,k rjk}. The existance of SR p,q (B) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of tranjugate lattice matrices and we show that a square lattice matrix can be expressed as meet (or greatest lower bound or infimum) of symmetric and tranjugate lattice matrices and we discuss their uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, G-majorization inequalities are proven for the gradients of two Gateaux differentiable G-increasing functions defined on a real linear space endowed with the structure of an Eaton triple. Such inequalities can be interpreted as a monotonicity property of certain operators and functionals with respect to a special preorder on the class of all G-increasing functions. The notion of a c-strongly G-increasing function is introduced and used to illustrate the derived inequalities. The obtained results are also applied for matrix Eaton triples connected with the eigenvalues map on the space of Hermitian matrices as well as with the singular values map on the space of complex matrices. [ABSTRACT FROM AUTHOR]

In this paper, we construct the spectral expansion for the nonself-adjoint differential operators generated in the space of vector functions by an ordinary differential expression of arbitrary order with periodic matrix coefficients, by using the essential spectral singularities, singular quasimomenta, and series with parenthesis. [ABSTRACT FROM AUTHOR]

The location of the roots of a quadratic scalar polynomial may be identified from its coefficients. This paper shows that when the coefficients of the polynomial are square matrices, then appropriate generalizations of some of these statements hold for the eigenvalues of the resulting quadratic matrix polynomial. The locations of the eigenvalues are described with respect to the imaginary axis, the unit circle or the real line. The results lead to upper bounds on some important distances associated with quadratic matrix polynomials. The principal tool used is an eigenvalue localization technique using block Geršgorin sets applied to certain linearizations of these polynomials that come from well known vector spaces. New bounds on the eigenvalues of the matrix polynomial arising from these localizations are also presented. [ABSTRACT FROM AUTHOR]

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable non‐planar curves of genus 0 and degree d$d$ in P3∖P1${\mathbb {P}}^3\backslash {\mathbb {P}}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non‐reductive Lie group. [ABSTRACT FROM AUTHOR]

Let H = (V,E) be a general hypergraph with rank m and co-rank mc and AH be its adjacency tensor, the p-spectral radius ρ(p)(H) of H is defined as ρ(p)(H)=maxx∈Sn-1p xT AHx, where Sn-1p = {x ∈ Rn| x |p=1}. For m = mc and p m, we know that there is a unique positive eigenvector x ∈ Sn-1p belonging to ρ(p)(H) and ρ(p)(H) can be computed by α -normal labeling method. In this paper, we generalize these properties to the case for m ≠ mc and some other properties are obtained. At the same time, some applications are also given on the properties attained in this paper. [ABSTRACT FROM AUTHOR]

For every m , n ∈ N and every field K , let M (m × n , K) be the vector space of the (m × n) -matrices over K and let S (n , K) be the vector space of the symmetric (n × n) -matrices over K. We say that an affine subspace S of M (m × n , K) or of S (n , K) has constant rank r if every matrix of S has rank r. Define A K (m × n ; r) = { S | S affine subspace of M (m × n , K) of constant rank r } A s y m K (n ; r) = { S | S affine subspace of S (n , K) of constant rank r } a K (m × n ; r) = max ⁡ { dim ⁡ S | S ∈ A K (m × n ; r) }. a s y m K (n ; r) = max ⁡ { dim ⁡ S | S ∈ A s y m K (n , r) }. In this paper we prove the following two formulas for r ≤ m ≤ n : a s y m R (n ; r) ≤ ⌊ r 2 ⌋ (n − ⌊ r 2 ⌋) a R (m × n ; r) = r (n − r) + r (r − 1) 2. [ABSTRACT FROM AUTHOR]

We construct stable vector bundles on the space P(SdCn+1) of symmetric forms of degree d in n + 1 variables which are equivariant for the action of SLn+1(C) and admit an equivariant free resolution of length 2. For n = 1, we obtain new examples of stable vector bundles of rank d - 1 on Pd, which are moreover equivariant for SL2(C). The presentation matrix of these bundles attains Westwick's upper bound for the dimension of vector spaces of matrices of constant rank and fixed size. [ABSTRACT FROM AUTHOR]

We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of C n in terms of the spectrum of both unperturbed and perturbed matrices as well as the spectrum of the unperturbed matrix only. The results presented give tighter bounds than the Davis–Kahan sinΘ theorem. We apply the result to a graph perturbation problem. [ABSTRACT FROM AUTHOR]

This article defines a distribution family based on the distribution of a random variable associated with the projections of a normally distributed random matrix with unknown mean onto a linear space and onto a closed convex cone. The stochastic monotonicity property of the distributions in the family is established. This property finds the application in the proving the unbiasedness of a likelihood ratio test in a multivariate order restricted model. [ABSTRACT FROM AUTHOR]

We classify all Rota–Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which do not arise from the decompositions of the entire algebra into a direct vector space sum of two subalgebras. [ABSTRACT FROM AUTHOR]

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ (n , K) \mathrm{GL}(n,K) , where 퐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ (n , K) \mathrm{SL}(n,K). Rosenberg described the normal subgroups of GL ⁢ (V) \mathrm{GL}(V) , where 푉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 푔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 푉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. [ABSTRACT FROM AUTHOR]

In earlier work, we showed how to handle the Group Theoretical issue of the Little Group for spin 1/2 tachyons by introducing a special metric in the vector space of one-particle states. Here that technique is extended to tachyons of any spin. Examining the bi-linear algebra of the generating matrices for spin 5/2, we find a complete basis for the Gell-Mann matrices that form the Lie algebra for SU(3). A Dirac-like equation is developed for tachyons of any integer-plus-one-half spin; and it shows multiple distinct mass eigenvalues. The primary model shows a mass spectrum (in the case of j = 5 / 2) that roughly mimics the known data on masses of the three neutrinos; the model can be tweaked to fit that experimental data precisely. [ABSTRACT FROM AUTHOR]

We investigate pattern-avoiding n × n (0 , 1) -matrices with emphasis on patterns of length 3: p q r -avoiding where { p , q , r } ⊆ { 1 , 2 , ... , n }. We show that all such maximal (0 , 1) -matrices contain the same number of 1's, and their structure is determined. We then show that the set of p q r -avoiding n × n permutation matrices span the linear space of dimension (n − 1) 2 + 1 generated by the n × n permutation matrices and determine a corresponding basis for each p , q , r. [ABSTRACT FROM AUTHOR]

Let m, n, p and q be positive integers such that p q ≤ m n and min (p , q) ≤ min (m , n). Let M m , n (C) denote the vector space of matrices of m rows and n columns with complex entries and let M n (C) stand for M n , n (C). For any matrix A ∈ M m (C) let σ (A) denote its spectrum. We prove that a couple of maps ϕ : M m , n (C) → M p , q (C) a n d ψ : M n , m (C) → M q , p (C) satisfy σ ϕ (M) ψ (N) = σ (M N) for all (M , N) ∈ M m , n (C) × M n , m (C) , if and only if there exist invertible matrices P ∈ M p (C) and Q ∈ M q (C) such that one of the following assertions holds: (m , n) = (p , q) , and we have ϕ (M) = P M Q and ψ (N) = Q − 1 N P − 1 for all (M , N) ∈ M m , n (C) × M n , m (C). (m , n) = (q , p) , and we have ϕ (M) = P M t Q and ψ (N) = Q − 1 N t P − 1 for all (M , N) ∈ M m , n (C) × M n , m (C). [ABSTRACT FROM AUTHOR]

Let V be a finite-dimensional vector space over the finite field F q and suppose W and W ˜ are subspaces of V. Two linear transformations T : W → V and T ˜ : W ˜ → V are said to be similar if there exists a linear isomorphism S : V → V with S W = W ˜ such that S ∘ T = T ˜ ∘ S W. Given a linear map T defined on a subspace W of V , we give an explicit formula for the number of linear maps that are similar to T. Our results extend a theorem of Philip Hall that settles the case W = V where the above problem is equivalent to counting the number of square matrices over F q in a conjugacy class. [ABSTRACT FROM AUTHOR]

We proposed a synonym search method to solve (A , B) ∼ (C , D) problem over time with a query by an example in a known domain for information in an unknown domain. It seems a natural relation that "Bush in the 2000s" is similar to "Reagan in the 1980s" because "Bush" and "Reagan" are the president of the USA in these decades. The abstraction is "A in B " which is similar to "C in D." We solve the (A , B) ∼ (C , D) problem over time by using transformation matrix word vectors over time in the Skip-gram model. The example of the (A , B) ∼ (C , D) problem is as below. For instance, the sentence "Bush in the 2000s" is similar to "X in the 1980s" is given. We search for the appropriate entity to X of the sentence. Therefore, we focus on the transitivity between the transformation matrix. Our approach is to convert the vector representation of "A " in the model of the word embedding model of the "B " to the vector representation of "X " in the model of the word embedding model of the "D " by getting the transformation matrix between word embedding models. We discuss the parameters of previous work and improve choosing words to make transformation matrix using co-occurrence cluster. Our aim is to search for synonyms in which there are more than the 1000 years of separation. However, there are a few common stable meaning words between the "2000s" and the "1000s." Therefore, in the situation, there is difficulty to use co-occurrence cluster because the clusters of common words are more than 100,000. That is why, we use the transitive relation (2000s, X) ∼ (1500s, Y) , (1500s, Y) ∼ (1000s, Z) ⇒ (2000s, X) ∼ (1000s, Z) (in the abstruction, the transitivity is x ∼ y , y ∼ z ⇒ x ∼ z ) to solve the (A , B) ∼ (C , D) problem over the 1000 years of separation. We had experiments as the demonstration to solve the (A , B) ∼ (C , D) problem and evaluate nDCG and MRR. [ABSTRACT FROM AUTHOR]

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see [12]. The proof involves elements from Integer Programming (totally unimodular systems of equations playing a role in particular) and employs Farkas' Lemma. The linear space of block upper triangular matrices can be viewed as being determined by a special pattern of zeros. The present paper is concerned with the question whether the decomposition result can be extended to situations where other, less restrictive, zero patterns play a role. It is shown that such generalizations do indeed hold for certain directed graphs determining the pattern of zeros. The graphs in question are what will be called L -free. This notion is akin to other graph theoretical concepts available in the literature, among them the one of being N -free in the sense of [16]. [ABSTRACT FROM AUTHOR]

Any square matrix over an algebraically closed field has a Jordan normal form. In this paper, we prove that every infinite upper triangular matrix over an arbitrary field has a generalized infinite Jordan normal form. [ABSTRACT FROM AUTHOR]

High-frequency link matrix rectifier (HFLMR) is advantageous in three-phase isolated ac–dc conversion due to its elimination of dc-link components. However, the input current distortion increases in an HFLMR when it operates in discontinuous conduction mode (DCM). The modulation scheme forms the high-frequency link current shape, which directly affects the input current distortion. Space vector pulsewidth modulation (SVPWM) is a popular modulation method for the matrix-type converter for its flexibility of arranging switching vectors. In this article, a new switching scheme based on SVPWM is proposed for the HFLMR to improve the current quality in DCM without adding complexity to the control. In DCM, the resonance between diode capacitance and transformer leakage inductance becomes a source of converter harmonics and instability. The method of shorting the transformer secondary is implemented to eliminate the voltage and current ringing in the discontinuous period. A 600-W lab prototype HFLMR that converts 115Vrms, 400-Hz three-phase input voltage to 270-V dc output voltage is implemented for experimental verification of the proposed switching scheme and converter topology. With the proposed switching scheme, the input current distortion is effectively reduced. With the proposed topology, the high-frequency voltage and current ringing are eliminated. [ABSTRACT FROM AUTHOR]

We study multivariate Gaussian models that are described by linear conditions on the concentration matrix. We compute the maximum likelihood (ML) degrees of these models. That is, we count the critical points of the likelihood function over a linear space of symmetric matrices. We obtain new formulae for the ML degree, one via line geometry, and another using Segre classes from intersection theory. We settle the case of codimension one models, and characterize the degenerate case when the ML degree is zero. [ABSTRACT FROM AUTHOR]

We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian. [ABSTRACT FROM AUTHOR]

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q -Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries. [ABSTRACT FROM AUTHOR]

In this paper, by using the matrix representation of generalized bicomplex numbers, we have defined the homothetic motions on some hypersurfaces in four dimensional generalized linear space R4α, β. Also, for some special cases we have given some examples of homothetic motions in R4 and R24 and obtained some rotational matrices, too. Therefore, we have investigated some applications about kinematics of generalized bicomplex numbers. [ABSTRACT FROM AUTHOR]

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A , B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t -tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver and an arbitrary finite-dimensional algebra, and so it is considered as hopeless. If (A , B) is such a pair, then Ker A ∩ Ker B ≠ 0. We give a simple normal form (A nor , B nor) of the matrices of (A , B) if Ker A ∩ Ker B is one-dimensional. We do not know whether it is canonical; i.e., whether (A nor , B nor) is uniquely determined by (A , B). We prove its uniqueness only if the Jordan canonical form of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic. The matrix A nor is the Weyr canonical form of A , and B nor commutes with A nor. In order to describe the structure of (A nor , B nor) , we describe explicitly all matrices commuting with a given Weyr matrix. [ABSTRACT FROM AUTHOR]

A variety of density matrix based methods for the analysis and visualization of electronic excitations are discussed and their implementation within the framework of the algebraic diagrammatic construction of the polarization propagator is reported. Their mathematical expressions are given and an extensive phenomenological discussion is provided to aid the interpretation of the results. Starting from several standard procedures, e.g., population analysis, natural orbital decomposition, and density plotting, we proceed to more advanced concepts of natural transition orbitals and attachment/ detachment densities. In addition, special focus is laid on information coded in the transition density matrix and its phenomenological analysis in terms of an electron-hole picture. Taking advantage of both the orbital and real space representations of the density matrices, the physical information in these analysis methods is outlined, and similarities and differences between the approaches are highlighted. Moreover, new analysis tools for excited states are introduced including state averaged natural transition orbitals, which give a compact description of a number of states simultaneously, and natural difference orbitals (defined as the eigenvectors of the difference density matrix), which reveal details about orbital relaxation effects. [ABSTRACT FROM AUTHOR]

We explore the relationship between local and global reducibility of spaces of nilpotent matrices. By local reducibility we mean that small subspaces of a given irreducible linear space L ⊆ M n (C) are reducible. One of our main results is that for certain integers m depending on n there is an (m + 1) -dimensional space L which is irreducible, but every one of its m -dimensional subspaces is, not just reducible, but simultaneously triangularizable. [ABSTRACT FROM AUTHOR]

We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directions are determined by integer vectors. We consider the two-dimensional case in detail, and also discuss the extension to three-dimensional vector spaces. [ABSTRACT FROM AUTHOR]

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from spin systems , or spin unitaries , which are finite sets S of selfadjoint unitary matrices such that any two unitaries in S anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all m -tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C ⁎ -envelope of any operator system generated by a spin system of cardinality 2 k or 2 k + 1 is the simple matrix algebra M 2 k (C). As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [13] that the spin ball B m spin and max ball B m max coincide as matrix convex sets in the cases m = 1 , 2. We also derive analogous results for countable spin systems and their C ⁎ -envelopes. [ABSTRACT FROM AUTHOR]

Background: Scales are linear combinations of variables with coefficients that add up to zero and have a similar meaning to "contrast" in the analysis of variance. Scales are necessary in order to incorporate genomic information into relationship matrices for genomic selection. Statistical and biological parameterizations using scales under different assumptions have been proposed to construct alternative genomic relationship matrices. Except for the natural and orthogonal interactions approach (NOIA) method, current methods to construct relationship matrices assume Hardy–Weinberg equilibrium (HWE). The objective of this paper is to apply vector algebra to center and scale relationship matrices under non-HWE conditions, including orthogonalization by the Gram-Schmidt process. Theory and methods: Vector space algebra provides an evaluation of current orthogonality between additive and dominance vectors of additive and dominance scales for each marker. Three alternative methods to center and scale additive and dominance relationship matrices based on the Gram-Schmidt process (GSP-A, GSP-D, and GSP-N) are proposed. GSP-A removes additive-dominance co-variation by first fitting the additive and then the dominance scales. GSP-D fits scales in the opposite order. We show that GSP-A is algebraically the same as the NOIA model. GSP-N orthonormalizes the additive and dominance scales that result from GSP-A. An example with genotype information on 32,645 single nucleotide polymorphisms from 903 Large-White × Landrace crossbred pigs is used to construct existing and newly proposed additive and dominance relationship matrices. Results: An exact test for departures from HWE showed that a majority of loci were not in HWE in crossbred pigs. All methods, except the one that assumes HWE, performed well to attain an average of diagonal elements equal to one and an average of off diagonal elements equal to zero. Variance component estimation for a recorded quantitative phenotype showed that orthogonal methods (NOIA, GSP-A, GSP-N) can adjust for the additive-dominance co-variation when estimating the additive genetic variance, whereas GSP-D does it when estimating dominance components. However, different methods to orthogonalize relationship matrices resulted in different proportions of additive and dominance components of variance. Conclusions: Vector space methodology can be applied to measure orthogonality between vectors of additive and dominance scales and to construct alternative orthogonal models such as GSP-A, GSP-D and an orthonormal model such as GSP-N. Under non-HWE conditions, GSP-A is algebraically the same as the previously developed NOIA model. [ABSTRACT FROM AUTHOR]

Two matrix vector spaces V , W ⊂ C n × n are said to be equivalent if S V R = W for some nonsingular S and R. These spaces are congruent if R = S T. We prove that if all matrices in V and W are symmetric, or all matrices in V and W are skew-symmetric, then V and W are congruent if and only if they are equivalent. Let F : U × ... × U → V and G : U ′ × ... × U ′ → V ′ be symmetric or skew-symmetric k -linear maps over C. If there exists a set of linear bijections φ 1 , ... , φ k : U → U ′ and ψ : V → V ′ that transforms F to G , then there exists such a set with φ 1 = ... = φ k. [ABSTRACT FROM AUTHOR]

Let V be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let [V] be the vector space of complex valued functions on V, and let [V]Z be the subgroup of [V] consisting of integer valued functions. We show that there exists a Z-basis of [V]Z consisting of characteristic functions of certain isotropic subspaces of V and such that the matrix of the Fourier transform from [V] to [V] with respect to this basis is triangular. We show that this is a special case of a result which holds for any two-sided cell in a Weyl group. [ABSTRACT FROM AUTHOR]

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We study actions of bosonizations of quantum linear spaces on quantum algebras. Under mild conditions, we classify actions on quantum affine spaces and quantum matrix algebras. In the former case, it is shown that all actions of generalized Taft algebras are trivial extensions of actions on quantum planes. In both cases we achieve bounds on the rank of the bosonization acting on the algebra. [ABSTRACT FROM AUTHOR]

Matrix codes over a finite field F q are linear codes defined as subspaces of the vector space of m × n matrices over F q . In this paper, we show how to obtain self-dual matrix codes from a self-dual matrix code of smaller size using a method we call the building-up construction. We show that every self-dual matrix code can be constructed using this building-up construction. Using this, we classify, that is, we find a complete set of representatives for the equivalence classes of self-dual matrix codes of small sizes. In particular we have classifications for self-dual matrix codes of sizes 2 × 4 , 2 × 5 over F 2 , of size 2 × 3 , 2 × 4 over F 4 , of size 2 × 2 , 2 × 3 over F 8 , and of size 2 × 2 , 2 × 3 over F 13 , all of which have been left open from K. Morrison's classification. [ABSTRACT FROM AUTHOR]