Group inverse for two classes of \(2\times 2\) block matrices over rings. (English) Zbl 1312.15004
The manuscript deals with the group inverse for particular classes of matrices over rings.
A square matrix \(A\) is said to be group invertible if there exists a matrix \(X\), denoted by \(X=A^{\sharp}\), such that \[ AXA=A, \;\;XAX=X \;\;\text{and} \;\;AX=XA. \]
In this paper, the authors obtain necessary and sufficient conditions for the existence of the group inverse of two classes of \(2 \times 2\) block matrices over an associative ring \(R\). In addition, the representations of the group inverse of two classes are given. The studied classes are the following:
A square matrix \(A\) is said to be group invertible if there exists a matrix \(X\), denoted by \(X=A^{\sharp}\), such that \[ AXA=A, \;\;XAX=X \;\;\text{and} \;\;AX=XA. \]
In this paper, the authors obtain necessary and sufficient conditions for the existence of the group inverse of two classes of \(2 \times 2\) block matrices over an associative ring \(R\). In addition, the representations of the group inverse of two classes are given. The studied classes are the following:
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- (1) \(M=\left( \begin{matrix} AX+YB & A \\ B & 0 \end{matrix} \right)\), where \(A, B, X, Y \in R^{n \times n}\), \(A^{\sharp}\) exists, \(XA=AX\) and \(X\) is invertible.
- ●
- (2) \(M=\left( \begin{matrix} A & B \\ C & D \end{matrix} \right)\), where \(A \in R^{n \times n}\), \(D \in R^{m \times m}\), \(CA=C\), \(AB=B\) and \((D-CB)^{\sharp}\) exists.
Reviewer: Juan Ramon Torregrosa Sanchez (Valencia)
MSC:
| 15A09 | Theory of matrix inversion and generalized inverses |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 16U30 | Divisibility, noncommutative UFDs |
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