In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-\lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${\mathbb R}^{n\times n}$, smoothly depending on parameters $x\in \Omega\subset {\mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

翻译：在这项工作中,我们根据两个参数来考虑对正确定铅笔。也就是说,我们所关注的是普遍的乙基价值问题$A(x)-lambda B(x)美元,其中美元和美元B(x)美元是美元对称矩阵价值的函数,在美元中,美元值和美元B(x)美元是对称矩阵值的值,在美元值和美元中,美元值值值值的价值值在美元中是相当的。我们首先对目前普遍存在的乙基值问题,从而对相应的预测,根据目前的乙基值问题,对乙基值的光度和源值的光度进行总体理论性分析,然后对铅笔的乙基值统计特性进行数字研究,其中美元值和美元值是完整的或带宽的,因为在某些参数值值值值值中(交汇点)是相等的。我们首先对目前普遍存在的乙基值和源值问题的光源值和源源值进行总体理论性研究,然后对铅笔的乙基值的统计特性进行数字性研究,然后对铅笔的精度价值进行数字性研究,因为美元和美元是完整的或带宽度,这是我们进行一个随机的基质研究。