arXiv subjects

Xuefeng Xu

Publications and source records attributed to Xuefeng Xu.

25 records · Page 2Linked to original sources

arXiv2017

New upper bounds for the spectral variation of a general matrix

Let $A\in\mathbb{C}^{n\times n}$ be a normal matrix with spectrum $\{\lambda_{i}\}_{i=1}^{n}$, and let $\widetilde{A}=A+E\in\mathbb{C}^{n\times n}$ be a perturbed matrix with spectrum $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$. If $\widetilde{A}$ is still normal, the celebrated Hoffman--Wielandt theorem states that there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that $\big(\sum_{i=1}^{n}|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}|^{2}\big)^{1/2}\leq\|E\|_{F}$, where $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix. This theorem reveals the strong stability of the spectrum of a normal matrix. However, if $A$ or $\widetilde{A}$ is non-normal, the Hoffman--Wielandt theorem does not hold in general. In this paper, we present new upper bounds for $\big(\sum_{i=1}^{n}|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}|^{2}\big)^{1/2}$, provided that both $A$ and $\widetilde{A}$ are general matrices. Some of our estimates improve or generalize the existing ones.

math.NA↗

arXiv2016

New perturbation bounds for the spectrum of a normal matrix

Let $A\in\mathbb{C}^{n\times n}$ and $\widetilde{A}\in\mathbb{C}^{n\times n}$ be two normal matrices with spectra $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$, respectively. The celebrated Hoffman--Wielandt theorem states that there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that $\left(\sum_{i=1}^{n}\big|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big|^{2}\right)^{1\over 2}$ is no larger than the Frobenius norm of $\widetilde{A}-A$. However, if either $A$ or $\widetilde{A}$ is non-normal, this result does not hold in general. In this paper, we present several novel upper bounds for $\left(\sum_{i=1}^{n}\big|\widetilde{\lambda}_{\pi(i)}-\lambda_{i}\big|^{2}\right)^{1\over 2}$, provided that $A$ is normal and $\widetilde{A}$ is arbitrary. Some of these estimates involving the "departure from normality" of $\widetilde{A}$ have generalized the Hoffman--Wielandt theorem. Furthermore, we give new perturbation bounds for the spectrum of a Hermitian matrix.

math.NA↗

arXiv2016

A new expression for the Moore-Penrose inverse of a class of matrices

An expression for the Moore-Penrose inverse of a matrix of the form M = XNY , where X and Y are nonsingular, has been recently established by Castro-Gonz\'alez et al. [1, Theorem 2.2]. The expression plays an essential role in developing explicit expressions for the Moore-Penrose inverse of a two-by-two block matrix. In this paper, we present a new expression for the Moore-Penrose inverse of this class of matrices, which improves the result in [1].

math.NA↗

arXiv2016

Generalization of the Sherman-Morrison-Woodbury formula involving the Schur complement

Let $X\in\mathbb{C}^{m\times m}$ and $Y\in\mathbb{C}^{n\times n}$ be nonsingular matrices, and let $N\in\mathbb{C}^{m\times n}$. Explicit expressions for the Moore-Penrose inverses of $M=XNY$ and a two-by-two block matrix, under appropriate conditions, have been established by Castro-Gonz\'{a}lez et al. [Linear Algebra Appl. 471 (2015) 353-368]. Based on these results, we derive a novel expression for the Moore-Penrose inverse of $A+UV^{\ast}$ under suitable conditions, where $A\in \mathbb{C}^{m\times n}$, $U\in \mathbb{C}^{m\times r}$, and $V\in \mathbb{C}^{n\times r}$. In particular, if both $A$ and $I+V^{\ast}A^{-1}U$ are nonsingular matrices, our expression reduces to the celebrated Sherman-Morrison-Woodbury formula. Moreover, we extend our results to the bounded linear operators case.

math.NA↗

arXiv2016

An improved upper bound for the number of distinct eigenvalues of a matrix after perturbation

An upper bound for the number of distinct eigenvalues of a perturbed matrix has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate is the central result in Farrell's work and can be applied to estimate the number of Krylov iterations required for solving a perturbed linear system. In this paper, we present an improved upper bound for the number of distinct eigenvalues of a matrix after perturbation. Furthermore, some results based on the improved estimate are presented.

math.NA↗

arXiv2015

Effects of Sliding Speed on the Intensity of Triboluminescence in Slide contact: Experimental Measurements and Theoretical Analyses

Triboluminescence (TL) is the emission of light produced by rubbing or striking two materials together. Here, the light emission has been observed from the sliding contact between two disks under dry condition. The effects of the sliding speed on the intensity of TL have been experimentally investigated. The results show that the intensity of the emission light increases significantly with the sliding speed. A theoretical model is also proposed and an analytical expression is deduced for the intensity of TL in the slide contact. The theoretical prediction is found consistent with the experimental results. The present work may be helpful to the understanding of the mechanism of light emission when friction.

cond-mat.mtrl-sci↗

arXiv2014

A combined field approach for the two-way coupling problem in the liquid evaporation

During liquid evaporation, the temperature of the liquid determines the saturated vapor pressure above it, which controls the evaporation rate and thus determines the liquid temperature through latent heat. Therefore, the equations for the vapor concentration in the atmosphere and for the temperature in the liquid are coupled and must be solved in an iterative manner. In the present paper, a combined field approach which unifies the coupled fields into one single field and thus makes the iteration unnecessary is proposed. The present work will be useful in scientific and industrial processes involving liquid evaporation and may also have more general applications to coupled field problems in which all the fields have the same governing equation.

cond-mat.soft↗