Thin Hessenberg pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V which satisfy both (i) and (ii) below.

(i)

There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^∗ is Hessenberg.

We call such a pair a thin Hessenberg pair (or TH pair). This is a special case of a Hessenberg pair which was introduced by the author in an earlier paper. We investigate several bases for V with respect to which the matrices representing A and A^∗ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an “oriented” version of called a TH system. We classify the TH systems up to isomorphism.     

A forbidden structure of ray nonsingular matrices

A complex square matrix is called a ray nonsingular matrix (RNS matrix) if its ray pattern implies that it is nonsingular. In this paper, a necessary condition for RNS matrices is provided by showing that if A=I−A(D)$A=I-A(D)$ is ray nonsingular, then the arc weighted digraph D contains no forbidden cycle chains.     

A quaternion QR algorithm

Summary This paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.     

On bilinear maps determined by rank one matrices with some applications

Let M_n be the algebra of all n×n matrix over a field F, A a rank one matrix in M_n. In this article it is shown that if a bilinear map ? from M_n×M_n to M_n satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from M_n to M_n such that . If ? is further assumed to be symmetric then there exists a matrix B such that ?(x,y)=tr(xy)B for all x,y∈M_n. Applying the main result we prove that if a linear map on M_n is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on M_n is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in M_n is an all-desirable point and an all-automorphisable point, respectively.     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Exact probabilities of correct classifications for uncorrelated repeated measurements from elliptically contoured distributions

Euclidean distance-based classification rules are derived within a certain nonclassical linear model approach and applied to elliptically contoured samples having a density generating function g. Then a geometric measure theoretical method to evaluate exact probabilities of correct classification for multivariate uncorrelated feature vectors is developed. When doing this one has to measure suitably defined sets with certain standardized measures. The geometric key point is that the intersection percentage functions of the areas under investigation coincide with those of certain parabolic cylinder type sets. The intersection percentage functions of the latter sets can be described as threefold integrals. It turns out that these intersection percentage functions yield simultaneously geometric representation formulae for the doubly noncentral g-generalized F-distributions. Hence, we get beyond new formulae for evaluating probabilities of correct classification new geometric representation formulae for the doubly noncentral g-generalized F-distributions. A numerical study concerning several aspects of evaluating both probabilities of correct classification and values of the doubly noncentral g-generalized F-distributions demonstrates the advantageous computational properties of the present new approach. This impression will be supported by comparison with the literature.It is shown that probabilities of correct classification depend on the parameters of the underlying sample distribution through a certain well-defined set of secondary parameters. If the underlying parameters are unknown, we propose to estimate probabilities of correct classification.     

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was:

Theorem. (See Borwein, Choi, Yazdani, 2001.) Letf(z)=±z±z²±?±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer then

     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

Nonnegative quadratic estimation and quadratic sufficiency in general linear models

Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for β^′Hβ+hσ² is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally, we derive a remark which indicates that our main results could be extended further to the quasi-normal case.     

Matrix multiplication is determined by orthogonality and trace

Any associative bilinear multiplication on the set of n-by-n matrices over some field of characteristic not two, that makes the same vectors orthogonal and has the same trace as ordinary matrix multiplication, must be ordinary matrix multiplication or its opposite.     

A mixed directed-undirected data structure for a parallel implementation of a domain decomposition algorithm

The choice of data structures influences the parallelization, efficiency and the manageability of a mesh refinement program. We introduce a mixed directed-undirected graph that combines both communication and scheduling needs. An inverted index is maintained for the directed graph to improve code performance and readability.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.     

A matrix problem over a central quadratic skew field extension

Let F⊂G=F(u) be a central quadratic skew field extension (such that the generator u is central in G) and a natural (G,G)-bimodule. We deal with the matrix problem on finding a canonical form for rectangular matrices over W with help of left elementary transformations of their rows and right elementary transformations of columns over G. We solve this problem reducing it in the separable (resp. inseparable) case to the semilinear (resp. pseudolinear) pencil problem.     

Schur product of matrices and numerical radius (range) preserving maps

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ° B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ° B) = F(?(A) ° ?(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.     

On the Kronecker Problem and related problems of Linear Algebra

We consider some classification problems of Linear Algebra related closely to the classical Kronecker Problem on pairs of linear maps between two finite-dimensional vector spaces. As shown by Djokovi? and Sergeichuk, the Kronecker’s solution is extended to the cases of pairs of semilinear maps and (more generally) pseudolinear bundles respectively. Our objective is to deal with the semilinear case of the Kronecker Problem, especially with its applications. It is given a new short solution both to this case and to its contragredient variant. The biquadratic matrix problem is investigated and reduced in the homogeneous case (in characteristic ≠2) to the semilinear Kronecker Problem. The integer matrix sequence Θ_n and Θ-transformation of polynomials are introduced and studied to get a simplified canonical form of indecomposables for the mentioned homogeneous problem. Some applications to the representation theory of posets with additional structures are presented.     

Infinite-dimensional triangularization

The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector space, without placing any restrictions on the dimension of the space or on the base field. We define a transformation T of a vector space V to be triangularizable if V has a well-ordered basis such that T sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that the following conditions (among others) are equivalent: (1) T is triangularizable, (2) every finite-dimensional subspace of V is annihilated by $f(T)$ for some polynomial f that factors into linear terms, (3) there is a maximal well-ordered set of subspaces of V that are invariant under T, (4) T can be put into a crude version of the Jordan canonical form. We also show that any finite collection of commuting triangularizable transformations is simultaneously triangularizable, we describe the closure of the set of triangularizable transformations in the standard topology on the algebra of all transformations of V, and we extend to transformations that satisfy a polynomial the classical fact that the double-centralizer of a matrix is the algebra generated by that matrix.     

Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, (d/dt)x(t)=Ax(t)$(\mathrm{d}/\mathrm{d}t)\mathbf{x}(t)=A\mathbf{x}(t)$, is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of (1/φ(h))(_xk+1-_xk)=A[θ_xk+1+(1-θ)_xk]$(1/\phi (h))({\mathbf{x}}_{k+1}-{\mathbf{x}}_k)=A[\theta {\mathbf{x}}_{k+1}+(1-\theta ){\mathbf{x}}_k]$; both derivatives are in the same form of (_xk+1-_xk)/φ(h)$(x_{k+1}-x_k)/\phi (h)$.     

Ideal Turaev-Viro invariants

Turaev-Viro invariants are defined via state sum polynomials associated to a special spine or a triangulation of a compact 3-manifold. By evaluation of the state sum at any solution of the so-called Biedenharn-Elliott equations, one obtains a homeomorphism invariant of the manifold (“numerical Turaev-Viro invariant”). The Biedenharn-Elliott equations define a polynomial ideal. The key observation of this paper is that the coset of the state sum polynomial with respect to that ideal is a homeomorphism invariant of the manifold (“ideal Turaev-Viro invariant”), stronger than the numerical Turaev-Viro invariants. Using computer algebra, we obtain computational results on several examples of ideal Turaev-Viro invariants, for all closed orientable irreducible manifolds of complexity at most 9.