Location of Ritz values in the numerical range of normal matrices

Majorization bounds for Ritz values of self-adjoint matrices

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the $\tan Θ$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values.

A note on a conjecture concerning rank one perturbations of singular M-matrices

A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.

通过广义特征值研究Sobolev正交多项式零点的有界性

Boundedness of zeros of Sobolev orthogonal polynomials via generalized eigenvalues

本文通过引入由无限Hermitian正定矩阵定义的矩阵Sobolev内积，利用广义特征值的渐近行为研究乘法算子有界性，从而给出Sobolev正交多项式零点集有界性的判定准则。