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On a solution of the quaternion matrix equation $X-A \widetilde{X} B=C$ and its application. (English) Zbl 1083.15019
The authors study the solvability and explicit representation by means of a characteristic polynomial of the solution to the matrix equation $X-AXB=C$. Then they study solvability of the matrix equation $(*)$ $X-A\widetilde{X}B=C$ where if $x=a+bi+cj+dk$ is a quaternion ($i^2=j^2=k^2=-1$, $ij=-ji=k$), then $\widetilde{x}=a-bi+cj-dk$. To solve equation $(*)$ explicitly a real representation of quaternion matrices is used. The results are applied to the resolution of the matrix equation $X-A\overline{X}B=C$ where $\overline{X}$ is the complex conjugate of $X$.

MSC:
15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
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References:
[1] Barnett, S., Storey, C.: Matrix Methods in Stability Theory, Nelson, London, 1970 · Zbl 0243.93017
[2] Barnett, S.: Matrices in Control Theory with Applications to Linear Programming, Van Nostrand Reinhold, New York, 1971 · Zbl 0245.93002
[3] Jameson, A.: Solution of the equation AX XB = C by inversion of an M {$\times$}M or N {$\times$} N matrix. SIAM J. Appl. Math., 16, 1020--1023 (1968) · Zbl 0169.35202 · doi:10.1137/0116083
[4] Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications, 2nd ed., Academic Press, New York, 1985 · Zbl 0558.15001