An alternative characterization of generalized projectors (Q1881057)

The authors introduce an alternative modification of a result of \textit{J. Groß} and \textit{G. Trenkler} [Linear Algebra Appl. 264, 463--474 (1997; Zbl 0887.15024)]. Their result asserts that for a square complex matrix \(K\) the following are equivalent. (i) \(K\) is a generalized projector, that is \(K^2=K^*\). (ii) \(K\) is quadripotent and normal, that is \(K^4=K\) and \(K^*K=KK^*\). (iii) \(K\) is quadripotent and partial isometry, that is \(K^4=K\) and \(KK^*K=K\), where \(K^*\) denotes the conjugate transpose of \(K\).