On linear combinations of generalized projectors Original Research Article

Baksalary and Baksalary [Linear Algebra Appl. 321 (2000) 3] established a complete solution to the problem of when a linear combination of two different projectors is also a projector by listing all situations in which nonzero complex numbers c1, c2 and nonzero complex matrices P1, P2 (P1≠P2) satisfying Pi2=Pi, i=1,2, form a matrix P=c1P1+c2P2 such that P2=P. In the present paper, the same problem is considered for generalized projectors G1 and G2 defined by Groß and Trenkler [Linear Algebra Appl. 264 (1997) 463] as matrices satisfying Gi2=Gi*, i=1,2. Their results concerning the sum G1+G2 and difference G1−G2 appear to be very special cases of the general solution established herein.