The invariance of elementary symmetric functions

The general problem considered is: what linear transformations on matrices preserve certain prescribed invariants or other properties of the matrices? Specifically, the forms of the following linear transformations are determined: the linear transformations that hold either the trace or the second elementary symmetric function of the eigenvalues of each matrix fixed, and in addition preserve either the determinant, or the permanent, or an elementary symmetric function of the squares of the singular values, or the property of being a rank 1 matrix or a unitary matrix.     

On the quadratic convergence of a generalization of the Jacobi Method to arbitrary matrices

A method of diagonalizing a general matrix is proved to be ultimately quadratically convergent for all normalizable matrices. The method is a slight modification of a method due to P. J. Eberlein, and it brings the general matrix into a normal one by a combination of unitary plane transformations and plane shears (non-unitary). The method is a generalization of the Jacobi Method: in the case of normal matrices it is equivalent to the method given by Goldstine and Horwitz.     

On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Summary A new class of elementary matrices is presented which are convenient in Jacobi-like diagonalisation methods for arbitrary real matrices. It is shown that the presented transformations possess the normreducing property and that they produce an ultimate quadratic convergence even in the case of complex eigenvalues. Finally, a quadratically convergent Jacobi-like algorithm for real matrices with complex eigenvalues is presented.     

幂幺矩阵的充要条件

研究幂幺矩阵的充要条件,利用矩阵的秩和齐次线性方程组解空间的维数.将m=2时幂幺矩阵的充要条件推广到一般幂幺矩阵的充要条件,得出了幂幺矩阵可对角化的结果,并将幂幺矩阵的充要条件平行地推广到幂幺线性变换.     

On the inertia and range of transformations of positive definite matrices

Results are established which relate the range and inertia of general transformations on positive definite matrices. Included are a bound theorem for certain eigenvalues of these transformations, a characterization of positive definite preserving, completely positive transformations, and generalizations of the theorems of Stein [8] and of Stein and Pfeffer [9].     

Monotone additive matrix transformations

We investigate additive transformations on the space of real or complex matrices that are monotone with respect to any admissible partial order relation. A complete characterization of these transformations is obtained. In the real case, we show that such transformations are linear and that all nonzero monotone transformations are bijective. As a corollary, we characterize all additive transformations that are monotone with respect to certain classical matrix order relations, in particular, with respect to the Drazin order, left and right *-orders, and the diamond order.     

Linear fractional transformations in rings and modules

This report examines the basic properties of linear fractional transformations of a matrix argument, with matrix coefficients, and still more general entities. The familiar properties of scalar transformations generalize surprisingly well: for instance, cross- ratios of matrices are preserved up to similarity. The questions of multiple transitivity and uniqueness of coefficients are examined, and two engineering applications are outlined.     

Rank one preservers between spaces of Boolean matrices

In this paper, we characterize (i) linear transformations from one space of Boolean matrices to another that send pairs of distinct rank one elements to pairs of distinct rank one elements and (ii) surjective mappings from one space of Boolean matrices to another that send rank one matrices to rank one matrices and preserve order relation in both directions. Both results are proved in a more general setting of tensor products of two Boolean vector spaces of arbitrary dimension.     

Finding Symmetry Groups of Some Quadratic Programming Problems

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help decrease the problem dimension, reduce the size of the search space by means of linear cuts. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms. We formulate conditions, which allow us to transform the original problem to a new system of coordinates, such that the symmetries may be sought only among orthogonal transformations. In particular, these conditions are satisfied if the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. We describe the structure and some useful properties of the group of symmetries of the problem. Besides that, the methods of detection of such symmetries are outlined for different special cases as well as for the general case.     

A new Jacobi-like method for joint diagonalization of arbitrary non-defective matrices

This paper addresses the problem of joint diagonalization of a set of matrices. A new Jacobi-Like method that has the advantages of computational efficiency and of generality is presented. The proposed algorithm brings the general matrices into normal ones and performs a joint diagonalization by a combination of unitary and shears (non-unitary) transformations. It is based on the iterative minimization of an appropriate cost function using generalized Jacobi rotation matrices.     

Some recent applications of functional equations to geometry

The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.     

On the simultaneous tridiagonalization of two symmetric matrices

We discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal–tridiagonal form under the very mild assumption that the matrix pencil is regular. We outline the general principles and propose a unified framework for the problem. This allows us to gain new insights, leading to an economical approach that only uses Gauss transformations and orthogonal Householder transformations. Numerical experiments show that the approach is numerically robust and competitive.     

Fast Givens rotations for orthogonal similarity transformations

Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.     

Linear transformations on matrices: rank preservers and determinant preservers (Note)

Characterizations of rank preserving and determinant preserving linear transformations on matrices over an algebraically closed field, due to Marcus and Frobenius respectively, are generalized to linear transformations on matrices over an arbitrary field (rank preservers) or over any field which satisfies a weak cardinality condition (determinant preservers).     

Christoffel Transforms and Hermitian Linear Functionals

In this contribution we are focused on some spectral transformations of Hermitian linear functionals. They are the analogues of the Christoffel transform for linear functionals, i. e. for Jacobi matrices which has been deeply studied in the past. We consider Hermitian linear functionals associated with a probability measure supported on the unit circle. In such a case we compare the Hessenberg matrices associated with such a probability measure and its Christoffel transform. In this way, almost unitary matrices appear. We obtain the deviation to the unit matrix both for principal submatrices and the complete matrices respectively.     

A generalization of the rotation matrix and related results

A new generalization of the rotation group involving a skew circulant matrix is given. Using the exponential map, a unified treatment is given to this generalization and to one due to Ungar. The special functions associated with the corresponding Lie groups are the trigonometric and hyperbolic functions of order n. Infinitesimal generators and invariants under the corresponding transformations are also obtained. A general theorem on linear transformations involving circulant and skew circulant matrices is also given.     

Classification of pairs consisting of a linear and a semilinear map

L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices [5, Chapter XII]. See also Dickson's paper [3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.     

Updating the QR decomposition of block tridiagonal and block Hessenberg matrices

We present an efficient block-wise update scheme for the QR decomposition of block tridiagonal and block Hessenberg matrices. For example, such matrices come up in generalizations of the Krylov space solvers MinRes, SymmLQ, GMRes, and QMR to block methods for linear systems of equations with multiple right-hand sides. In the non-block case it is very efficient (and, in fact, standard) to use Givens rotations for these QR decompositions. Normally, the same approach is also used with column-wise updates in the block case. However, we show that, even for small block sizes, block-wise updates using (in general, complex) Householder reflections instead of Givens rotations are far more efficient in this case, in particular if the unitary transformations that incorporate the reflections determined by a whole block are computed explicitly. Naturally, the bigger the block size the bigger the savings. We discuss the somewhat complicated algorithmic details of this block-wise update, and present numerical experiments on accuracy and timing for the various options (Givens vs. Householder, block-wise vs. column-wise update, explicit vs. implicit computation of unitary transformations). Our treatment allows variable block sizes and can be adapted to block Hessenberg matrices that do not have the special structure encountered in the above mentioned block Krylov space solvers.     

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.     

三维Mbius变换中的Beardon定理

本文利用高维保向Ｍｏｂｉｕｓ变换的Ｃｌｉｆｆｏｒｄ矩阵表示，对三维广义椭圆Ｍｏｂｉｕｓ群，椭圆Ｍｏｂｉｕｓ群及运动Ｍｏｂｉｕｓ群进行了讨论，得到了它们各自的特征。