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Full investigation of the matrix equation $ AX+XB=C$ and specifically of the equation $ AX-XA=C$


Author: E. L. Rabkin
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 26 (2014), nomer 1.
Journal: St. Petersburg Math. J. 26 (2015), 117-130
MSC (2010): Primary 15A24
DOI: https://doi.org/10.1090/S1061-0022-2014-01333-7
Published electronically: November 21, 2014
MathSciNet review: 3234807
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Abstract: In the paper, the matrix equations $ AX-XA=C$ and $ AX+XB=C$ (the Lyapunov equation) are fully investigated and solved if a solution exists. As special cases, exact expressions for the resolvent of the equation $ (E-A)X=0$ are obtained for a finite-dimensional operator $ A$ and the Fredholm equation of the second kind is studied completely in the finite-dimensional case. The form of a matrix $ C$ with nonnegative entries is found such that it is the commutator of a given matrix $ A$ with nonnegative entries and some other matrix $ X$ with nonnegative entries.


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Additional Information

E. L. Rabkin
Affiliation: Division of Mathematics, Bonch-Bruevich St. Petersburg State University of Telecommunications, 22 pr. Bol′shevikov, St. Petersburg, Russia
Email: rabk@sut.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01333-7
Keywords: Matrix, commutator, Lyapunov equation
Received by editor(s): October 1, 2012
Published electronically: November 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society