Nonsingularity of the difference and the sum of two idempotent matrices

作者：Zuo Kezheng^*

来源：Linear Algebra and Its Applications, 2010, 433(2): 476-482.

DOI：10.1016/j.laa.2010.03.013

• Idempotent matrix
• Direct sum
• Rank of matrix

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).

• 出版日期2010-8-1
• 单位湖北师范大学