论文标题
任意铅笔特征结构的对角线尺度
Diagonal scalings for the eigenstructure of arbitrary pencils
论文作者
论文摘要
在本文中，我们展示了如何为任意矩阵铅笔构建对角线尺度$λb-a $，其中$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a。这种对角线尺度的目的是在某种意义上“平衡”铅笔的行和列规范。我们看到，缩放矩阵铅笔的问题等同于缩放特定非负矩阵的行和列总和的问题。但是，众所周知，存在无法任意缩放的正方形和非负矩阵。为了解决此问题，我们考虑了一个近似嵌入式问题，其中相应的非负矩阵为正方形，可以始终缩放。然后，新的缩放方法基于用于缩放平方非阴性矩阵的sindhorn-knopp算法，并具有全部支持，以使其具有双重随机或变体。此外，使用U. G. Rothblum和H. Schneider（1989）的结果，我们为存在平方非负矩阵的对角线尺度的零模式提供了简单的条件，以使行和列和列和列和列的任何处方公共矢量。我们从数值上说明铅笔的新缩放技术提高了其特征值计算的准确性。
In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $λB-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that can not be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn-Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider (1989), we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues.
