Sufficient Conditions on Commutators for a Pair of Stable Matrices to Have a Common Solution to the Lyapunov Equation

Authors:
Thomas J. Laffey;Helena Šmigoc
Affiliations:
thomas.laffey@ucd.ie and helena.smigoc@fmf.uni-lj.si;-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2010

Citing 6
Cited 1

Topics in matrix analysis

Topics in matrix analysis
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Original article: A new type of recurrent neural networks for real-time solution of Lyapunov equation with time-varying coefficient matrices

Mathematics and Computers in Simulation

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Abstract

This paper presents sufficient conditions on the commutators of two matrices $A$ and $B$ for the existence of a nonsingular matrix $T$, such that matrices $TAT^{-1}$ and $TBT^{-1}$ are in upper block triangular form, with blocks of matching sizes on the diagonal, where the sizes of diagonal blocks are less than or equal to $2\times2$. With this result sufficient conditions for the existence of a common solution to the Lyapunov equation for two matrices $A$ and $B$ with $\mathrm{rank}(A-B)=2$ are derived.