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The group inverse of the combinations of two idempotent matrices. (English) Zbl 1220.15008

The authors analyze the group inverse of \[ aP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQ \] of two different idempotent matrices \(P\) and \(Q\), where \(a, b, c, d, e, f, g \in \mathbb{C}\) and \(a \neq 0\), \(b \neq 0\). They derive its explicit expressions in two cases: \(a+b+c+d+e+f+g \neq 0\) and \(a+b+c+d+e+f+g = 0\), under the conditions \((PQ)^2=(QP)^2\) and \((PQ)^2=0\) or \((QP)^2=0\).
The authors also obtain some necessary and sufficient conditions for the existence of the group inverse of \(aP+bQ+cPQ\) and discuss its explicit expressions.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A27 Commutativity of matrices
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