Diagonal scalings for the eigenstructure of arbitrary pencils
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In this paper we show how to construct diagonal scalings for arbitrary matrix pencils λ 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 method is then based on the Sinkhorn-Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic. In addition, using results of U. G. Rothblum and H. Schneider (1989), we give sufficient conditions for the existence of diagonal scalings of square nonnegative matrices to be not only doubly stochastic but have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the sensitivity of the computation of their eigenvalues.
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