Roth's solvability criteria for the matrix equations AX−XˆB=C and X−AXˆB=C over the skew field of quaternions with an involutive automorphism q↦qˆ

By Vyacheslav Futorny, T. Klymchuk, V. Sergeichuk. 2016

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Abstract The matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X − A X B = C over a field has a solution if and only if the matrices [ A C 0 I ] and [ I 0 0 B ] are simultaneously equivalent to [ A 0 0 I ] and [ I 0 0 B ] . We extend these criteria to the matrix equations A X − X ˆ B = C and X − A X ˆ B = C over the skew field of quaternions with a fixed involutive automorphism q ↦ q ˆ .

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

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