论文标题
在分离良好的矩阵的扰动上
On the perturbations of well separated matrices
论文作者
论文摘要
如果其所有Gershgorin圆圈都远离单位圆并彼此分开，则矩阵分开。在本文中，特征值中的相对误差区域是作为二次椭圆形的，用于非对角线分离矩阵的非对角线扰动。因此给出了可计算的相对误差，该误差在Gershgorin圆圈参数方面绑定。当分离为$ o（n）$并且矩阵是正定的，则会提出一个扰动下特征值的交织定理。此外，当分离为$ O（n^2）$时，特征向量矩阵的条件号被上限以获得扰动特征值的区域。数值结果表明，即使矩阵分离不太分开，对角线条目与特征向量入口的大小之间的关系。我们利用这种趋势在使用功率方法估算Perron载体方面。
A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non diagonal perturbation of well seperated matrices. Thus giving a computable relative error bound in terms of Gershgorin circle parameters. When the separation is $O(n)$ and the matrix is positive definite, an interlacing theorem for the eigenvalues under perturbation is presented. Further when the separation is $O(n^2 )$, condition number of the eigenvector matrix is upper bounded to obtain the region of perturbed eigenvalue. Numerical results show the relation between diagonal entries and the magnitude of the eigenvector entries even when the matrix is not so well separated. We exploit this trend in estimating the Perron vector using power method.
