Idempotency of the Hermitian part of a complex matrix Original Research Article

In 1980, Khatri (Linear Alg. Appl. 33 (1980) 57–65) has shown that the Hermitian part (A +A*)/2 of a square complex matrixA is idempotent and has the same rank asA if and only ifA is normal and the real part of any of its non-trivial eigenvalues is equal to one. In this note we investigate idempotency of (A +A*)/2 without assumptions on its rank and demonstrate thatA is necessarily an EP matrix, whereA is called EP if the range ofA equals the range ofA*.