Nonsingularity of linear combinationsof idempotent matrices Original Research Article

Groß and Trenkler [SIAM J. Matrix Anal. Appl. 21 (1999) 390] pointed out that if a difference of idempotent matrices P1 and P2 is nonsingular, then so is their sum, and Koliha et al. [Linear Algebra Appl., in press] expressed explicitly a condition, which combined with the nonsingularity of P1+P2 ensures the nonsingularity of P1−P2. In the present note, these results are strengthened by showing that the nonsingularity of P1+P2 is in fact equivalent to the nonsingularity of any linear combination c1P1+c2P2, wherein c1+c2≠0. Some other nonsingularity-type relationships referring to linear combinations of P1 and P2 are also established.