Nonsingularity of the difference and the sum of two idempotent matrices

摘要

Gro beta and Trenkler [1] pointed out that if a difference of idempotent matrices P and Q is nonsingular, then so is their sum, and Koliha et al. [2] expressed explicitly a condition, which combined with the nonsingularity of P + Q ensures the nonsingularity of P - Q. In the present paper, these results are strengthened by showing that the nonsingularity of P - Q is in fact equivalent to the nonsingularity of any combination aP + bQ - cPQ (where a not equal 0, b not equal 0, a + b = c), and the nonsingularity of P + Q is equivalent to the nonsingularity of any combination aP + bQ - cPQ (where a not equal 0, b not equal, a + b not equal c).