In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the ta...In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .展开更多
Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-...Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （z））, and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein＇s equation and its generalization which naturally leads to a certain recursive equation.展开更多
文摘In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .
基金Much of the work was done when the author was visiting the Department of Mathematics, Harvard University, under the project from the Y. C. Tang Disciplinary Development Fund, Zhejiang University. The author thanks Professor H. T. Yau and Professor S. T. Yau for their hospitality during the visit. The referees＇ careful reading helps to improve the presentation of the paper. This work was Partially supported by the National Natural Science Foundation of China （Grant No. 11071213）, the Natural Science Foundation of Zhejiang Province （No. R6090034）, and the Doctoral Program ~and of Ministry of Education （No. J20110031）.
文摘Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （z））, and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein＇s equation and its generalization which naturally leads to a certain recursive equation.
