关于极分解和广义极分解的一些新结果

本文研究极分解和广义极分解.孙和陈提出的Frobenius范数下的逼近定理被推广至任何酉不变范数情形.得到了次酉极因子的一个新的表达式.通过新的表达式,我们得到了次酉极因子在任何酉不变范数下的扰动界.最后,讨论了数值计算方法.     

Rank Reducing Matrix Norms

We consider approximation numbers for some norms on matrices, and look at the question when a closest rank h p approximant can be chosen to reduce the rank of a matrix by p . If the latter is always possible, we call the norm rank p reducing. It is easily seen that any unitarily invariant norm is rank p reducing. We show that any absolute norm on $\shadC^{n \times m}$ is rank n m 1 reducing and that the numerical radius norm on $ \shadC^{n\times n}$ is rank n m 1 reducing as well. Non-examples and computations of approximation numbers are also presented.     

The Chebyshev approximation of a rectangular matrix by matrices of smaller rank as the limit of lp-apprpoximation

In this paper we investigate a connection between l_p-approximation and the Chebyshev approximation of a rectangular matrix by matrices of smaller rank. We consider also the stationary points of problems (4) and (5) which are connected with these approximations.     

Interpolating inequalities related to a recent result of Audenaert

Audenaert recently obtained an inequality for unitarily invariant norms that interpolates between the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality for matrices. A refined version of Audenaert’s inequality for the Hilbert–Schmidt norm is given. Other interpolating inequalities for unitarily invariant norms are also presented.     

Some New Perturbation Bounds for Subunitary Polar Factors

In this paper, we present some new perturbation bounds for subunitary polar factors in a special unitarily invariant norm called a Q-norm. Some recent results in the Frobenius norm and the spectral norm are extended to the Q-norm on one hand. On the other hand we also present some relative perturbation bounds for subunitary polar factors.     

Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25 :362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A ? Ã‖₂ ? ‖A ? Ã‖_F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14 :588–593) for both real and complex square matrices are generalized. Copyright © 2005 John Wiley & Sons, Ltd.     

Norms and inequalities for condition numbers,III

The main results provide comparisons between condition numbers (based on unitarily invariant norms) of (i) positive definite (Hermitian) matrices A, B and of A + B, (ii) a positive definite matrix and its principal submatrix, and (iii) a matrix and an augmented form of the matrix.     

Approximation by Partial Isometries and Symmetric Approximation of Finite Frames

We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize all the solutions. In particular, this allows us to give a simple necessary and sufficient condition for uniqueness. We then apply these results to solve the global problem of approximation by partial isometries, and to extend the notion of symmetric approximation of frames introduced in Frank et al. (Trans Am Math Soc 354: 777–793, 2002). In addition, we characterize symmetric approximations of frames belonging to a prescribed subspace.     

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded almost surely. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type.     

Perturbation analysis for best approximation and the polar factor by subunitary matrices

This paper is a continuation and improvement over the results of Laszkiewicz and Zietak [BIT, 2006, 46: 345–366], studying perturbation analysis for polar decomposition. Some basic properties of best approximation subunitary matrices are investigated in detail. The perturbation bounds of the polar factor are also derived.      

On the existence of unitarily invariant norm under some conditions

Let r ₁, …, r _m be positive real numbers and A ₁, …, A _m be n × n matrices with complex entries. In this article, we present a necessary and sufficient condition for the existence of a unitarily invariant norm ‖·‖, such that ‖A _i ‖ = r _i , for i = 1, …, m. Then we identify the greatest unitarily invariant norm which satisfies this condition. Using this, we get an approximation of unitarily invariant norms. Although the minimum unitarily invariant norm which satisfies this condition does not exist in general, we find conditions over A _i s and r _i s which are sufficient for the existence of such a norm. Finally, we get a characterization of unitarily invariant norms.     

Matrix semigroups determined by unitarily invariant norms

The purpose of this paper is to study the structure of the matrix semigroups defined by unitarily invariant norms and, equivalently, those defined by arbitrary ellipsoidal norms. Among other things it is found that when an element of such a semigroup has a semi-inverse, the semi-inverse is unique, and, in the case of unitarily invariant norms, this is the Moore-Penrose generalized inverse. The symmetric gauge functions that determine submultiplicative matrix norms are characterized, and these norms are related to the spectral norm.     

Norm inequalities for matrix geometric means of positive definite matrices

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.     

矩阵的酉不变范数不等式

利用凹函数和半正定矩阵的性质,讨论并且得到了一些矩阵Rotfel型范数不等式.另外,通过研究Hermitian矩阵和斜Hermitian矩阵和的特征值的模行列式的不等式,得到一些关于Hermitian矩阵和斜Hermitian矩阵和的范数不等式.推广了文献中的相关结果.     

Norm equalities and inequalities for operator matrices

Several norm equalities and inequalities for operator matrices are proved in this paper. These results, which depend on the structure of circulant and skew circulant operator matrices, include pinching type inequalities for weakly unitarily invariant norms.     

Separation theorems for singular values of matrices and their applications in multivariate analysis

Separation theorems for singular values of a matrix, similar to the Poincaré separation theorem for the eigenvalues of a Hermitian matrix, are proved. The results are applied to problems in approximating a given r.v. by an r.v. in a specified class. In particular, problems of canonical correlations, reduced rank regression, fitting an orthogonal random variable (r.v.) to a given r.v., and estimation of residuals in the Gauss-Markoff model are discussed. In each case, a solution is obtained by minimizing a suitable norm. In some cases a common solution is shown to minimize a wide class of norms known as unitarily invariant norms introduced by von Neumann.     

Inequalities for hadamard product and unitarily invariant norms of matrices

The paper contains some general theorems for Hadamard product of matrices which in particular include Fiedler's Theorem and a better bound for an inequality on product of eigenvalues of certain matrices due to Ando. Lieb's concavity Theorem has been proved using operator means. Some inequalities for unitarily invariant norms have also been proved.     

Weighted norms in subspace‐based methods for time series analysis

Many modern approaches of time series analysis belong to the class of methods based on approximating high‐dimensional spaces by low‐dimensional subspaces. A typical method would embed a given time series into a structured matrix and find a low‐dimensional approximation to this structured matrix. The purpose of this paper is twofold: (i) to establish a correspondence between a class of SVD‐compatible matrix norms on the space of Hankel matrices and weighted vector norms (and provide methods to construct this correspondence) and (ii) to motivate the importance of this for problems in time series analysis. Examples are provided to demonstrate the merits of judiciously selecting weights on imputing missing data and forecasting in time series. Copyright © 2016 John Wiley & Sons, Ltd.     

Chapter 5: linear preservers on functions of Singular values

Many authors have studied linear preservers on functions of singular values. If the functions are suitably chosen, the problem reduces to the characterization of linear isometrics of unitarily invariant norms on matrices. In this chapter we survey the results on these subjects.     

Linear groups of degree n containing an element with exactly n–2 equal eigenvalues

Many authors have studied linear preservers on functions of singular values. If the functions are suitably chosen, the problem reduces to the characterization of linear isometrics of unitarily invariant norms on matrices. In this chapter we survey the results on these subjects.