On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

A nonsymmetric matrix with integer eigenvalues

A nonsymmetric N?×?N matrix with elements as certain simple functions of N distinct real or complex numbers r ₁, r ₂, …, r_N is presented. The matrix is special due to its eigenvalues???the consecutive integers 0,1,2, …, N?1. Theorems are given establishing explicit expressions of the right and left eigenvectors and formulas for recursive calculation of the right eigenvectors. A special case of the matrix has appeared in sampling theory where its right eigenvectors, if properly normalized, give the inclusion probabilities of the conditional Poisson sampling design.     

Dimension of quasicircles

We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala’s conjecture that the Hausdorff dimension of a k-quasicircle is at most 1+k ².     

A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues

This paper describes a new computational procedure for calculating eigenvalues and eigenvectors of a square matrix. The method is based on a matrix function, the sign of a matrix. Eigenvalues and eigenvectors of matrices with distinct eigenvalues and nondefective matrices with repeated roots can be determined in a straightforward manner. Defective matrices require additional calculations.     

The sign matrix and the separation of matrix eigenvalues

The sign matrices uniquely associated with the matrices (M ? ζ_jI)², where ζ_j are the corners of a rectangle oriented at π/4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.     

A method for separating nearly multiple eigenvalues for Hermitian matrix

In this paper, we propose a numerical method to verify for nearly multiple eigenvalues of a Hermitian matrix not being strictly multiple eigenvalues. From approximate eigenvalues computed, it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones, and if they are very close each other, the verification method for simple eigenvalues may fail to enclose them separately, because of singularity of the system in the verification. There are several methods for enclosing multiple and nearly multiple eigenvalues (e.g., [Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl. 324 (2001) 209–226]), For such cases, there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple, up to now. So, for enclosed eigenvalues, we propose a numerical method to separate nearly multiple eigenvalues.     

Normal and seminormal eigenvalues of matrix functions

LetP() be ann×n analytic matrix function andW(P) be its numerical range. In this paper classical results on the normality of matrix eigenvalues on W(P) are generalized to the context of such matrix functions. Special attention is paid to corners of W(P) and to the special case of matrix polynomialsP().     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

A note on estimating eigenvalues of scale matrix of the multivariate F-distribution

Let F _pxp have the multivariate F-distribution with a scale matrix and degrees of freedom n ₁and n ₂. In this paper the problem of estimating eigenvalues of is considered. By constructing the improved orthogonally invariant estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqqHuoaraSqabeaacaqGEbaaaOGaaiikaiaadAeacaGG% Paaaaa!402A!\[\mathop \Delta \limits^{\rm{\^}} (F)\] of , which are analogous to Haff-type estimators of a normal covariance matrix, new estimators of eigenvalues of are given. This is because the eigenvalues of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqqHuoaraSqabeaacaqGEbaaaOGaaiikaiaadAeacaGG% Paaaaa!402A!\[\mathop \Delta \limits^{\rm{\^}} (F)\] are taken as estimates of the eigenvalues of .     

Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix

Several relative eigenvalue condition numbers that exploit tridiagonal form are derived. Some of them use triangular factorizations instead of the matrix entries and so they shed light on when eigenvalues are less sensitive to perturbations of factored forms than to perturbations of the matrix entries. A novel empirical condition number is used to show when perturbations are so large that the eigenvalue response is not linear. Some interesting examples are examined in detail.     

A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix

A fast method for computing all the eigenvalues of a Hamiltonian matrix M is given. The method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties. The algorithm requires about one-fourth the number of floating-point operations and one-half the space of the standard QR algorithm. The computed eigenvalues are shown to be the exact eigenvalues of a matrix M + E where ∥E∥ depends on the square root of the machine precision. The accuracy of a computed eigenvalue depends on both its condition and its magnitude, larger eigenvalues typically being more accurate.     

Computation of the small eigenvalues of a matrix

We present the ALGOL programdeigv which computes the small (in absolute value) isolated eigenvalues and their eigenvectors for a real diagonalizable matrix. Test samples are included.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 80, pp. 167–180, 1978.     

A Gerschgorin theorem for linear difference equations and eigenvalues of matrix products

The Gerschgorin circle theorem is used here to give sufficient conditions for the solution space of the difference equation x(m+1) = A (m+1)x(m) to admit a type of exponential dichotomy. The result obtained is then used to establish a result on regions of eigenvalue inclusion for the product of finitely many square matrices. An application to differential equations is also given.     

The eigenvalues of a tridiagonal matrix in biogeography

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory.     

Function of a square matrix with repeated eigenvalues

An analytical function f(A) of an arbitrary n×n constant matrix A is determined and expressed by the “fundamental formula”, the linear combination of constituent matrices. The constituent matrices Z_kh, which depend on A but not on the function f(s), are computed from the given matrix A, that may have repeated eigenvalues. The associated companion matrix C and Jordan matrix J are then expressed when all the eigenvalues with multiplicities are known. Several other related matrices, such as Vandermonde matrix V, modal matrix W, Krylov matrix K and their inverses, are also derived and depicted as in a 2-D or 3-D mapping diagram. The constituent matrices Z_kh of A are thus obtained by these matrices through similarity matrix transformations. Alternatively, efficient and direct approaches for Z_kh can be found by the linear combination of matrices, that may be further simplified by writing them in “super column matrix” forms. Finally, a typical example is provided to show the merit of several approaches for the constituent matrices of a given matrix A.     

拟圆周的两个几何性质

§1　IntroductionLetΓbe a Jordan curve of R2 and f∶R2→R2 be a k-quasiconformal mapping,where1≤k<+∞.Γis called a quasicirlce ifΓis the image of the unit circle B2 under f.It is well-known that quasicircles play a very important role in quasiconformalmapping theory,complex dynamics,Fuchsian groups,Teichmuller space theory and lowdimensional topology,( see[1—5] etc.)In1 963 ,Ahlfors obtained the three-point property of quasidisks[6] .Later,Gehring[7] ,Osgood[8] ,Krzyz[9] ,Ch…     

矩阵特征值的一类新的包含域

用盖尔圆盘定理来估计矩阵的特征值是一个经典的方法,这种方法仅利用矩阵的元素来确定特征值的分布区域.本文利用相似矩阵有相同的特征值这一理论,得到了矩阵特征值的一类新的包含域,它们与盖尔圆盘等方法结合起来能提高估计的精确度.