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PreprintCC BY 4.0arXiv

Conditioning of solutions to the Sylvester equation

  • Massimiliano Fasi
  • Behnam Hashemi
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Abstract

We partially answer an open problem, posed by Nick Higham, concerning the conditioning of solutions to Sylvester and Lyapunov equations. The question arises in the backward stability analysis of numerical algorithms for these equations. We first show that the solution to the Sylvester equation AX−XB = C can be arbitrarily ill-conditioned even if $A, B, C$ and the Kronecker sum I ⊗ A − BT ⊗ I are all perfectly conditioned. We then derive general a priori bounds on the condition number of the solution, as well as bounds for the Sylvester equation when A and B are diagonalizable. We also provide lower bounds involving matrix exponentials and Zolotarev numbers. For the Lyapunov equation AX+XAT = −C, we obtain upper bounds in two settings: (i) when A is symmetric positive definite while C is symmetric negative definite, and (ii) when A is strictly dissipative and C is symmetric positive definite.

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Cite

BibTeXRIS
BibTeX
@misc{fasi2026conditioning,
  title = {{Conditioning of solutions to the Sylvester equation}},
  author = {Massimiliano Fasi and Behnam Hashemi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04050},
  eprint = {2609.04050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04050},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Conditioning of solutions to the Sylvester equation
AU  - Massimiliano Fasi
AU  - Behnam Hashemi
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04050
UR  - https://arxiv.org/abs/2609.04050
AB  - We partially answer an open problem, posed by Nick Higham, concerning the conditioning of solutions to Sylvester and Lyapunov equations. The question arises in the backward stability analysis of numerical algorithms for these equations. We first show that the solution to the Sylvester equation AX−XB = C can be arbitrarily ill-conditioned even if $A, B, C$ and the Kronecker sum I ⊗ A − B^T ⊗ I are all perfectly conditioned. We then derive general a priori bounds on the condition number of the solution, as well as bounds for the Sylvester equation when A and B are diagonalizable. We also provide lower bounds involving matrix exponentials and Zolotarev numbers. For the Lyapunov equation AX+XA^T = −C, we obtain upper bounds in two settings: (i) when A is symmetric positive definite while C is symmetric negative definite, and (ii) when A is strictly dissipative and C is symmetric positive definite.
ER  -