Full investigation of the matrix equation $AX+XB=C$ and specifically of the equation $AX-XA=C$
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E. L. Rabkin
Translated by: N. B. Lebedinskaya - St. Petersburg Math. J. 26 (2015), 117-130
- DOI: https://doi.org/10.1090/S1061-0022-2014-01333-7
- Published electronically: November 21, 2014
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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.References
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Bibliographic 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
- Received by editor(s): October 1, 2012
- Published electronically: November 21, 2014
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 26 (2015), 117-130
- MSC (2010): Primary 15A24
- DOI: https://doi.org/10.1090/S1061-0022-2014-01333-7
- MathSciNet review: 3234807