Roth's solvability criteria for the matrix equations and over the skew field of quaternions with an involutive automorphism

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• 作者：Vyacheslav Futorny^a ; ^{futorny@ime.usp.br" class="auth_mail" title="E-mail the corresponding author} ; Tetiana Klymchuk^b ; ^{tetiana.klymchuk@upc.edu" class="auth_mail" title="E-mail the corresponding author} ; Vladimir V. Sergeichuk^c ; ^{sergeich@imath.kiev.ua" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A24 ; 15B33
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：510
• 期：Complete
• 页码：246-258
• 全文大小：323 K

文摘

The matrix equation AX−XB=C$AX-XB=C$ has a solution if and only if the matrices $\left[\begin{array}{cc}\hfill A\hfill & \hfill C\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right]$ and $\left[\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right]$ 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−AXB=C$X-AXB=C$ over a field has a solution if and only if the matrices $\left[\begin{array}{cc}\hfill A\hfill & \hfill C\hfill \\ \hfill 0\hfill & \hfill I\hfill \end{array}\right]$ and ecbce5cb75b3cc9df6a5a1884d">$\left[\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right]$ are simultaneously equivalent to $\left[\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill I\hfill \end{array}\right]$ and ecbce5cb75b3cc9df6a5a1884d">$\left[\begin{array}{cc}\hfill I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right]$. We extend these criteria to the matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$ over the skew field of quaternions with a fixed involutive automorphism $q\mapsto \hat{q}$.