Idempotency of the Hermitian part of a complex matrix (Q1300825)

The paper characterizes matrices \(A\) with the Hermitian part, \(H(A)= (A + A^*)/2,\) being idempotent. It is shown that such matrices belong to the class of complex \(m\times m\) matrices for which \(A A^+ = A^+ A,\) where \(A^+\) is the Moore-Penrose inverse of \(A\), or equivalently, the range of \(A\) equals the range of \(A^*\).