强P除环上方阵的酉相似理论(Ⅱ)

This is a continuation of the previous paper ( 1 ) . In this paper , a useful basic theorem that every selfconjugate matrix over the strong p division ring Ω is unitary similar to a tridiagorial matrix over the conter of Ω is given thus all of famous results involving selfconjugate matrices, positivedefinite selfcon jugate matrices, nonnegative selfconjugate matrix in the ordiniry com plex matrix theory are generalized to selfconjugate matrices over Ω . and Sigular decomuposition as well as polar decomposition in the ordinary complex matrix theory are also generalized to matrices over Ω .     

强p除环上方阵的酉相似理论(Ⅲ)

In this third paper, the famous Schur theorem that an n× n complex matrix is unitary similar to an upper triangular matrix is generalized to the socalled ∑-lizable matrix over the strong p-division ring Ω, where ∑ is the algebraically closed extension field of the center of Ω , and ∑?Ω.The generalized Schur's identity and other results involving the general-ized normal matrix over Ω is obtained by using this generalized Schur theorem.     

Schur定理在四元数体上的推广

In this paper，we show that every matrix over the real quaternion division ring is unitary similar to an upper triangular matrix.     

加权极分解

In this paper, a new matrix decomposition called the weighted polar decomposition is considered. Two uniqueness theorems of weighted polar decomposition are presented, and the best approximation property of weighted unitary polar factor and perturbation bounds for weighted polar decomposition are also studied.     

中心代数上一矩阵方程的中心对称与中心斜对称解

Let Ω be a finite dimensional central algebra and chart Ω≠2 .The matrix equation AXB-CXD=E over Ω is considered.Necessary and sufficient conditions for the existence of centro(skew)symmetric solutions of the matrix equation are given.As a particular case ,the matrix equation X-AXB=C over Ω is also considered.     

Wavelets and Wavelet Packets Related to a Class of Dilation Matrices

The construction of wavelets generated from an orthogonal multiresolution analysis can be reduced to the unitary extension of a matrix, which is not easy in most cases. Jia and Micchelli gave a solution to the problem in the case where the dilation matrix is 21 and the dimension does not exceed 3. In this paper, by the method of unitary extension of a matrix, we obtain the construction of wavelets and wavelet oackets related to a class of dilation matrices.     

A new algorithm for pole assignment of single-input linear systems using state feedback

In this paper we present a new algorithm for the single-input pole assignment problem using state feedback. This algorithm is based on the Schur decomposition of the closed-loop system matrix, and the numerically stable unitary transformations are used whenever possible, and hence it is numerically reliable.The good numerical behavior of this algorithm is also illustrated by numerical examples.     

Least-Squares Solution with the Minimum-Norm for the Matrix Equation A^TXB ＋ B^TX^TA = D and Its Applications

An efficient method based on the projection theorem,the generalized singular value decompositionand the canonical correlation decomposition is presented to find the least-squares solution with the minimum-norm for the matrix equation A~TXB B~TX~TA=D.Analytical solution to the matrix equation is also derived.Furthermore,we apply this result to determine the least-squares symmetric and sub-antisymmetric solution ofthe matrix equation C~TXC=D with minimum-norm.Finally,some numerical results are reported to supportthe theories established in this paper.     

Hlder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for Hlder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in R2n , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Hlder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Hlder exponents, the diameter of Γ and a specific d-sum (d d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.     

MINIMIZATION PROBLEM FOR SYMMETRIC ORTHOGONAL ANTI-SYMMETRIC MATRICES

By applying the generalized singular value decomposition and the canonical correlation decomposition simultaneously, we derive an analytical expression of the optimal approximate solution ^-X, which is both a least-squares symmetric orthogonal anti-symmetric solu- tion of the matrix equation A^TXA = B and a best approximation to a given matrix X^＊. Moreover, a numerical algorithm for finding this optimal approximate solution is described in detail, and a numerical example is presented to show the validity of our algorithm.     

RECURSIVE REPRODUCING KERNELS HILBERT SPACES USING THE THEORY OF POWER KERNELS

The main objective of this work is to decompose orthogonally the reproducing kernels Hilbert space using any conditionally positive definite kernels into smaller ones by introducing the theory of power kernels, and to show how to do this decomposition recursively. It may be used to split large interpolation problems into smaller ones with different kernels which are related to the original kernels. To reach this objective, we will reconstruct the reproducing kernels Hilbert space for the normalized and the extended kernels and give the recursive algorithm of this decomposition.     

The Haar Wavelet Analysis of Matrices and Its Applications

It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert time-domain problems into frequency-domain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function,a Haar-like wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an m × n matrix, the computational complexity is O(mn). In addition,when the method is applied to k-means clustering, one can obtain that k-means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems.In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results.     

Rankin-Cohen Deformations and Representation Theory

The author uses the unitary representation theory of SL2（R） to understand the Rankin-Cohen brackets for modular forms. Then this interpretation is used to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established.     

四元数矩阵的奇异值分解及其应用

In this paper, a constructive proof of singular value decomposition of quaternion matrix is given by using the complex representation and companion vector of quaternion matrix and the computational method is described. As an application of the singular value decomposition, the CS decomposition is proved and the canonical angles on subspaces of Q^n is studied.     

矩阵方程ATXA=B的对称正交对称解及其最佳逼近

By applying the generalized singular value decomposition of matrices, this paper provides the necessary and sufficient conditions for the existence and the expression of the symmetric ortho-symmetric solutions of the linear matrix equation A^TXA = B. In addition, the expression of the optimal approximation solution to the given matrix is derived.     

ERROR ANALYSIS FOR A FAST NUMERICAL METHOD TO A BOUNDARY INTEGRAL EQUATION OF THE FIRST KIND

For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [J. Comput. Math., 22 （2004）, pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.     

对称矩阵的β-性质及其Scaling稳定性分析

In this paper, a concept of the β-property of symmetric matrices is presented which is useful in the perturbation theory for matrices. A necessary and sufficient condition for a symmetric matrix to have the β-property, and the constant β,when it exists, are given. Further, the scaling stability of the symmetric matrix which has the β-property is investigated.     

Transition distributions of young diagrams under periodically weighted plancherel measures

Kerov[16,17] proved that Wigner＇s semi-circular law in Gauss[an unitary ensembles is the transition distribution of the omega curve discovered by Vershik and Kerov[34] for the limit shape of random partitions under the Plancherel measure. This establishes a close link between random Plancherel partitions and Gauss[an unitary ensembles, In this paper we aim to consider a general problem, namely, to characterize the transition distribution of the limit shape of random Young diagrams under Poissonized Plancherel measures in a periodic potential, which naturally arises in Nekrasov＇s partition functions and is further studied by Nekrasov and Okounkov[25] and Okounkov[28,29]. We also find an associated matrix mode[ for this transition distribution. Our argument is based on a purely geometric analysis on the relation between matrix models and SeibergWitten differentials.     

ON BLOCK MATRICES ASSOCIATED WITH DISCRETE TRIGONOMETRIC TRANSFORMS AND THEIR USE IN THE THEORY OF WAVE PROPAGATION

Block matrices associated with discrete Trigonometric transforms （DTT＇s） arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT＇s are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are unitary and offer simultaneous diagonalizations of specific matrix algebras. The main tool for the investigation of the aforementioned wave applications is the efficient inversion of such types of block matrices. To this direction, in this paper we develop an efficient algorithm for the inversion of matrices with U-diagonalizable blocks （U a fixed unitary matrix） by utilizing the U- diagonalization of each block and subsequently a similarity transformation procedure. We determine the developed method＇s computational complexity and point out its high efficiency compared to standard inversion techniques. An implementation of the algorithm in Matlab is given. Several numerical results are presented demonstrating the CPU-time efficiency and accuracy for ill-conditioned matrices of the method. The investigated matrices stem from real-world wave propagation applications.     

Higher Level Orderings on Modules

The aim of this paper is to investigate higher level orderings on modules over commutative rings. On the basis of the theory of higher level orderings on fields and commutative rings, some results involving existence of higher level orderings are generalized to the category of modules over commutative rings. Moreover, a strict intersection theorem for higher level orderings on modules is established.