上三角矩阵代数上的可加保极小秩映射

The additive mappings that preserve the minimal rank on the algebra of all n × n upper triangular matrices over a field of characteristic 0 are characterized.     

Similarity transformations of decomposable matrix polynomials and related questions

A relationship is found between the similarity transformations of decomposable matrix polynomials with relatively prime elementary divisors and the equivalence transformations of the corresponding matrices with scalar entries. Matrices with scalar entries are classified with respect to equivalence transformations based on direct sums of lower triangular almost Toeplitz matrices. This solves the similarity problem for a special class of finite matrix sets over the field of complex numbers. Eventually, this problem reduces to the one of special diagonal equivalence between matrices. Invariants of this equivalence are found.     

Tridiagonal and upper triangular inverse M-matrices

In this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse M-matrices. We give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class. This equivalence is extended to include some reducible matrices.     

Analytic Classification of Fuchsian Singular Points

We study the local analytic classification of Fuchsian singular points. The resonance formal normal form (FNF) of a system with a Fuchsian singular point, as well as the local analytic equivalence of a system to its resonance FNF, is well known. However, there are distinct resonance FNFs locally analytically equivalent to each other. The main theorem of the paper reduces the problem of local analytic equivalence of resonance FNFs to a problem about conjugacy of certain matrices associated to two FNFs (which are nil-triangular) by a block upper triangular matrix. As a consequence, the local analytic classification of Fuchsian singular points reduces to the study of the orbits of the group of block upper triangular matrices on nil-triangular matrices by conjugation.     

The structure and geometry of 4×4 pandiagonal matrices

The 4×4 pandiagonal matrices are tesselated by a group of transformations with two generators, which are analogous to a rotation and a rotatory inversion, acting on a single vector. These matrices have equivalence classes that are tesselated by a subgroup associated with a triangular tesselation of the sphere. The relationship of various subgroups to the permutation structure of pandiagonal matrices is studied.     

Some linear preserver problems on upper triangular matrices

We consider linear preserver problems on the algebra of upper triangular matrices. We obtain the general form of nonsingular linear maps preserving rank one idempotents or commutativity. We give examples to show that the nonsingularity assumption is essential in our results.     

Some linear preserver problems on upper triangular matrices

We consider linear preserver problems on the algebra of upper triangular matrices. We obtain the general form of nonsingular linear maps preserving rank one idempotents or commutativity. We give examples to show that the nonsingularity assumption is essential in our results.     

范德蒙型方程组最小二乘解的快速算法

在用多项式进行曲线拟合等实际问题中,需要求解以范德蒙型矩阵VT为系数阵的线性方程组VTx=b的最小二乘解.     

On some factorizations of two-parameter polynomial matrices

This paper is a logical continuation of the author's discussion about the solution of spectral problems for two-parameter polynomial matrices of general type. Various rank factorization algorithms are suggested, among them the so-called minimal factorization of a singular two-parameter polynomial matrix of degenerate rank into a product of some matrices whose ranks are equal to the rank of the original matrix. Spectral properties of these matrices are studied. The notion of minimal factorization is also extended to one-parameter polynomial and constant matrices. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 94–116 Translated by V. N. Kublanovskaya.     

Bases of spaces of matrices satisfying rank conditions

Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases for such spaces of maximum dimension can be chosen in such a way to consist of matrices of the minimal rank. An estimate of the ranks of matrices in particular types of bases for maximal such spaces is also given. This extends previously known results which were valid only in the case of spaces consisting of matrices of rank not equal to one.     

Positive matrix semigroups with binary diagonals

We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in S{\mathcal{S}} satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.     

Bounds for condition numbers of triangular and trapezoid matrices

Lower and upper bounds are determined for the conditioning of triangular and trapezoid matrices of full rank. These bounds can justify column pivoting in Gaussian elimination and in orthogonal elimination, or symmetrical pivoting in Cholesky factorization, or singular value decomposition instead of elimination.     

Three linear preservers on infinite triangular matrices

We consider three linear preserver problems on the algebra of infinite triangular matrices over fields. We characterize the maps preserving invertible and noninvertible matrices, the surjective maps preserving inverses and the surjective maps preserving rank permutability.     

Modular Automorphisms of Upper Triangular Matrices over a Commutative Ring Preserving Rank One

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The elementary divisor theory over the Hurwitz order of integral quaternions

There exists a diagonal form with certain divisibility conditions for matrices over the Hurwitz order of integral quaternions under unimodular equivalence. The diagonal entries are uniquely determined up to similarity. Given two such diagonal forms, where the diagonal entries are similar by pairs, the matrices prove to be ummodularly equivalent, whenever the rank of the matrices is creater than one.     

Minimal rank completions of partial banded matrices

It is proven that the minimal rank of a partial banded matrix equals the maximum of the minimal ranks of all triangular subpatterns. This proves partly the minimal rank conjecture in a paper by N. Cohen, C. R. Johnson, L. Rodman and H. J. Woerdeman (Operator Theory: Advances and Applications 40 (1989), 165-185).

The results are applied to the problem of simultaneously completing a matrix and its inverse.     

A classification of upper equivalent matrices the generic case

Two square matricesA andB are called upper equivalent iff there exists an invertible lower triangular matrixL such thatL ^–1 AL andB have the same upper triangular parts. In this paper we obtain a set of invariants for this equivalence relation. In the case when any minor of a matrixA, in the intersection of its last columns and first rows and contained in its upper triangular part is different from zero, it is shown that the above mentioned set of invariants completely determines the upper triangular parts of the matrices from the equivalence class ofA. Simple representatives for this class are also given.     

Additive mappings on symmetric matrices

Additive mappings, which do not increase the minimal rank of symmetric matrices are classified in characteristic two or three.     

Atomic representations of rank 2 graph algebras

We provide a detailed analysis of atomic ∗-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal ∗-dilations of defect free representations modelled on finite groups of rank 2.     

The ubiquity of free subsemigroups of infinite triangular matrices

In this note we show, that in a suitable sense, almost all k-generator subsemigroups of the multiplicative semigroup of infinite upper triangular matrices over a finite field are free of rank k.