Nick Higham’s open problem on conditioning of Sylvester and Lyapunov equation solutions

Given matrices $A$ , $B$ , and $C$ , determine conditions under which the solution $X$ of the Sylvester equation $AX-XB=C$ is well-conditioned, rather than merely existing uniquely or being nonsingular. Likewise, determine conditions under which the solution $X$ of the Lyapunov equation $AX+XA^T=-C$ is well-conditioned. The general question includes identifying when conditioning of the equation data and of the associated Kronecker operator $I\otimes A-B^T\otimes I$ does, or does not, control the conditioning of the solution.

References

Primary source

arXiv

Additional references

Conditioning of solutions to the Sylvester equation — arXiv

matrix-analysis numerical-linear-algebra

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new manuscript reports substantial partial progress on the problem but does not settle the general question.

Nick Higham’s problem asks for conditions under which solutions of Sylvester and Lyapunov equations are well conditioned, rather than merely nonsingular. The broader question remains unresolved.

Known results

Earlier perturbation theory gives controllability-based conditions guaranteeing nonsingularity, but not well conditioning; the Lyapunov case is more difficult because of its extra structure.

September 2026 partial answer

A new manuscript claims a negative result for general conditioning, together with a priori and diagonalizable-case bounds and lower bounds using matrix exponentials and Zolotarev numbers. It clarifies why conditioning of the computed solution cannot be inferred from conditioning of the equation data alone, but the broader problem is not completely resolved; this reported advance remains unverified.

Current status (as of September 2026): General conditioning remains open, while the new manuscript claims significant bounds and a negative result for overly general conditioning statements.

Sources

No solutions have been posted yet.