Balancing regular matrix pencils

Lemonnier, D;Van Dooren, Paul
(2006) SIAM Journal on Matrix Analysis and Applications — Vol. 28, n° 1, p. 253-263 (2006)

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  • Lemonnier, D
    Author
  • Van Dooren, PaulUCLouvain
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Abstract
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.
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Lemonnier, D., & Van Dooren, P. (2006). Balancing regular matrix pencils. SIAM Journal on Matrix Analysis and Applications, 28(1), 253-263. https://doi.org/10.1137/S0895479804440931 (Original work published 2006)