The nullity and rank of linear combinations of idempotent matrices

✍ Scribed by J.J. Koliha; V. Rakočević

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 88 KB
• Volume: 418
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2006.01.011

✦ Synopsis

Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004) 25-29] proved that the nonsingularity of P 1 + P 2 , where P 1 and P 2 are idempotent matrices, is equivalent to the nonsingularity of any linear combinations c 1 P 1 + c 2 P 2 , where c 1 , c 2 / = 0 and c 1 + c 2 / = 0. In the present note this result is strengthened by showing that the nullity and rank of c 1 P 1 + c 2 P 2 are constant. Furthermore, a simple proof of the rank formula of Groß and Trenkler [J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999) 390-395] is obtained.