Bounds for the Singular Values of a Matrix

A lower-bound theorem is developed for the singular values ofa matrix A (and therefore for the eigenvalues of B = A^HA). Itis found that there is always a unique column scaling of A whichproduces the optimum bound. However, sharper bounds still maysometimes be obtained by taking advantage of matrix partitioning.It is shown that the resulting bounds may often (but not always)be better than those obtained by applying Gerschgorin's theoremto B. The equivalent upper-bound theorem is found to be weak.     

矩阵的广义奇异值的若干不等式

本文引进广义奇异值概念，获得了若干广义奇异值不等式及交错定理．     

矩阵的奇异值与特征值的关系探究及应用

讨论矩阵的奇异值与特征值的关系,并给出奇异值在最优化理论分析中的一个应用.     

四元数矩阵的特征值与奇异值估计

In this paper, we give accurate estimation of eigenvalues and singular values of A + B,C^*AC and AB, where A, B and C are quaternions matrices. These results improve and generalze the results in [4] and [5]. We also obtainsum (?),for k=1,…,n. Where A and B are self-conjugate quaternions matrices of order n, and λ₁≥…≥λ_n,μ₁≥μ_n,λ₁,(A + B)≥…≥λ_n(A+B) be the eigenvalues of A,B and A + B, respectively.     

求矩阵奇异值分解的一种新方法

旨在给出求矩阵奇异值分解的一种新方法.改进和克服了以往方法的缺陷和不足.     

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.     

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m ( n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.     

Guaranteed Error Bounds for Eigenvalues of Singular Sturm‐Liouville Problems

We develop a simple oscillation theory for singular Sturm‐Liouville problems and combine it with recent asymptotic results, and with the AWA interval‐arithmetic code for integration of initial value problems with guaranteed error bounds, to obtain eigenvalue approximations with guaranteed error bounds for a class of singular Sturm‐Liouville problems. We believe that this is the first time that this has been achieved for singular eigenvalue problems.     

Hermite矩阵特征值的新扰动界

本文研究了Hermite矩阵特征值的任意扰动,给出了新的绝对和相对扰动界．所给出的界改进了Hoffman-Wielandt和Kahan早期的结果．     

Exploiting Mixed Precision for Computing Eigenvalues of Symmetric Matrices and Singular Values

The use of higher precision preconditioning for the symmetric eigenvalue problem and the singular value problem of general non–structured non–graded matrices are discussed. The matrix Q from the QR–decomposition as a preconditioner, applied to A with higher precision, in combination with Jacobi's method seems to allow the computation of all eigenvalues of symmetric positive definite matrices rsp. all singular values of general matrices to nearly full accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounds on Singular Values Revealed by QR Factorizations

We introduce a pair of dual concepts: pivoted blocks and reverse pivoted blocks. These blocks are the outcome of a special column pivoting strategy in QR factorization. Our main result is that under such a column pivoting strategy, the QR factorization of a given matrix can give tight estimates of any two a priori-chosen consecutive singular values of that matrix. In particular, a rank-revealing QR factorization is guaranteed when the two chosen consecutive singular values straddle a gap in the singular value spectrum that gives rise to the rank degeneracy of the given matrix. The pivoting strategy, called cyclic pivoting, can be viewed as a generalization of Golub's column pivoting and Stewart's reverse column pivoting. Numerical experiments confirm the tight estimates that our theory asserts.     

四元数长方矩阵乘积的奇异值估计

本文获得四元数长方矩阵乘积的奇异值的一些估计，改进了近期的一些结果．     

Traces of Singular Values and Borcherds Products

Let p be a prime for which the congruence group ₀(p)* is ofgenus zero, and let be the corresponding Hauptmodul. Let f be a nearly holomorphic modularform of weight 1/2 on ₀(4p) which satisfies some congruencecondition on its Fourier coefficients. We interpret f as a vectorvalued modular form. Applying Borcherds lifting of vector valuedmodular forms we construct infinite products associated to and relate them to Zagier's traceformula for the singular values of . 2000 Mathematics Subject Classification 11F03, 11F30 (primary);11F22, 11F37, 11F50 (secondary).     

Bounds on Norms of Compound Matrices and on Products of Eigenvalues

An upper bound on operator norms of compound matrices is presented,and special cases that involve the l₁, l₂ and l norms are investigated.The results are then used to obtain bounds on products of thelargest or smallest eigenvalues of a matrix. 1991 MathematicsSubject Classification 15A15, 15A18, 15A42.     

算子奇异值与迹的一些不等式

本文给出了算子奇异值与迹的一些不等式，推广了［１］的结果．     

Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G‐functions, or equivalently hypergeometric functions , also referred to as hyper‐Bessel functions. In the case it is well known that the corresponding gap probability for no squared singular values in (0, s) can be evaluated in terms of a solution of a particular sigma form of the Painlevé III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalized this formalism to general , but has not exhibited its reduction. After detailing the necessary working in the case , we consider the problem of reducing the 12 coupled differential equations in the case to a single differential equation for the resolvent. An explicit fourth‐order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third‐order nonlinear equation. The small and large s asymptotics of the fourth‐order equation are discussed, as is a possible relationship of the systems to so‐called four‐dimensional Painlevé‐type equations.     

Some Simple Estimates for the Singular Values of Matrices

Abstract We first provide a simple estimate for ||A~(-1)||_∞ and ||A~(-1)||_1 of a strictly diagonally dominant matrixA. On the Basis of the result, we obtain an estimate for the smallest singular value of A. Secondly, by scalingwith a positive diagonal matrix D, we obtain some simple estimates for the smallest singular value of an H-matrix, which is not necessarily positive definite. Finally, we give some examples to show the effectiveness ofthe new bounds.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.     

双调和算子本征值的上界

设Ω是 Rn中的有界区域 ,其边界足够光滑 ,λk为双调和算子在自由边界条件下的第 k个本征值 ,利用变分原理及 Fourier变换 ,给出了本征值部分和 ∑kj=1λj的一个上界 ,该上界仅依赖于区域的体积 .