Balancing Regular Matrix Pencils
Damien Lemonnier, Paul Van Dooren · SIAM Journal on Matrix Analysis and Applications · 2006
We present a new diagonal balancing technique for regular matrix pencils $\lambda B-A$, which aims at reducing the sensitivity of the corresponding generalized eigenvalues. It is inspired by the balancing technique of a square matrix A and has a comparable complexity. The diagonally scaled pencil has row and column norms that are balanced in a precise sense. We also show that balancing a pencil boils down to making it closer to some standardized normal pencil. We give numerical examples illustrating that the sensitivity of generalized eigenvalues of a pencil may significantly improve after balancing.