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Zbl 1204.15005
Ali Khan, Israr; Wang, Qing-Wen; Song, Guang-Jing
Minimal ranks of some quaternion matrix expressions with applications. (English)
[J] Appl. Math. Comput. 217, No. 5, 2031-2040 (2010). ISSN 0096-3003

Let $p(X, Y) = A - BX - X^{(*)}B^{(*)} - CYC^{(*)}$ and $q(X, Y) = A - BX + X^{(*)}B^{(*)} - CYC^{(*)}$ be quaternion matrix expressions, where $A$ is persymmetric or perskew-symmetric. The authors derive the minimal rank formula of $p(X, Y)$ with respect to pair of matrices $X$ and $Y = Y^{(*)}$, and the minimal rank formula of $q(X, Y)$ with respect to pair of matrices $X$ and $Y = - Y^{(*)}$. As applications, they establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.
[A. Arvanitoyeorgos (Patras)]

MSC 2000:: *15A03 Vector spaces
15A33 Matrices over special rings
15A24 Matrix equations

Keywords: minimal rank; linear matrix expression; persymmetric solution; quaternion matrix; linear quaternion matrix equations

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Zentralblatt MATH Berlin [Germany]