In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, 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.
翻译:在本文中,我们展示了如何为任意的矩阵铅笔($\lambda B-A$)构建对称缩放比例,其中,美元和美元都是复杂的基质(平方或非平方)。这种对称缩放的目的是在某种意义上平衡铅笔的行和柱规范。我们看到,缩放一个矩阵铅笔的问题相当于一个特定的非负式矩阵的行和列总和的缩放问题。然而,众所周知,存在着不能任意缩放的正方和非方非正方非正方矩阵。为了解决这一问题,我们考虑了一个大致嵌入的问题,其中相应的非负式矩阵是正方形的,而且总是可以缩放。然后,新的缩放方法基于Sinkhorn-Knopp算法,以平方非正方形的基质缩放比例,同时支持更明显地调整某个非正向性矩阵的大小。此外,我们使用U.G.Rothblum和H.Schneider(1989)的结果,在零模式上给出了无法任意缩放的充足条件,用以显示其平面的平面的平面的平面的平面的平面图。