Liu, Xiaoji; Wu, Lingling; Yu, Yaoming The group inverse of the combinations of two idempotent matrices. (English) Zbl 1220.15008 Linear Multilinear Algebra 59, No. 1-3, 101-115 (2011). 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. Reviewer: Juan Ramon Torregrosa Sanchez (Valencia) Cited in 1 ReviewCited in 15 Documents MSC: 15A09 Theory of matrix inversion and generalized inverses 15A27 Commutativity of matrices Keywords:group inverse; idempotent matrix; linear combination PDFBibTeX XMLCite \textit{X. Liu} et al., Linear Multilinear Algebra 59, No. 1--3, 101--115 (2011; Zbl 1220.15008) Full Text: DOI References: [1] DOI: 10.1016/S0024-3795(00)00225-1 · Zbl 0984.15021 · doi:10.1016/S0024-3795(00)00225-1 [2] DOI: 10.1016/j.laa.2003.09.008 · Zbl 1081.15016 · doi:10.1016/j.laa.2003.09.008 [3] DOI: 10.1016/j.laa.2004.01.011 · Zbl 1057.15018 · doi:10.1016/j.laa.2004.01.011 [4] DOI: 10.1016/S0024-3795(02)00343-9 · Zbl 1016.15027 · doi:10.1016/S0024-3795(02)00343-9 [5] DOI: 10.1016/j.laa.2007.02.016 · Zbl 1119.15025 · doi:10.1016/j.laa.2007.02.016 [6] Ben-Israel A, Generalized Inverses: Theory and Applications,, 2. ed. (2003) [7] DOI: 10.1016/j.laa.2005.02.027 · Zbl 1077.15022 · doi:10.1016/j.laa.2005.02.027 [8] DOI: 10.1080/03081080701535872 · Zbl 1163.15028 · doi:10.1080/03081080701535872 [9] Chen Y, Acta Math. Sin. 50 pp 1171– (2007) [10] DOI: 10.1090/S0002-9939-05-08091-3 · Zbl 1089.47002 · doi:10.1090/S0002-9939-05-08091-3 [11] DOI: 10.1016/j.laa.2006.01.011 · Zbl 1104.15001 · doi:10.1016/j.laa.2006.01.011 [12] Koliha JJ, Integral Equ. Oper. Theory 99 pp 1– (2006) [13] DOI: 10.1016/j.laa.2004.03.008 · Zbl 1060.15011 · doi:10.1016/j.laa.2004.03.008 [14] DOI: 10.1016/j.amc.2003.10.027 · Zbl 1070.15009 · doi:10.1016/j.amc.2003.10.027 [15] DOI: 10.1016/j.amc.2007.11.019 · Zbl 1165.15022 · doi:10.1016/j.amc.2007.11.019 [16] Zhang S, Math. FA 1406 pp 1– (2009) [17] Zuo K, J. Math. (PRC) 29 pp 285– (2009) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.