Diagonal Scalings for the Eigenstructure of Arbitrary Pencils

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 is “balance” some sense the row column norms pencil. We see that problem scaling a pencil equivalent sums particular nonnegative matrix. However, it known there exist square nonsquare cannot be scaled arbitrarily. To address issue, consider an approximate embedded problem, corresponding can always scaled. new methods then based on Sinkhorn--Knopp algorithm with total support doubly stochastic variant it. addition, using results U. G. Rothblum H. Schneider [Linear Algebra Appl., 114--115 (1989), pp. 737--764], give simple sufficient conditions zero pattern existence have any prescribed common vector sums. illustrate numerically techniques improve accuracy computation their eigenvalues.