The nullity and rank of linear combinations of idempotent matrices

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 comb...

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Bibliographic Details
Published in:Linear algebra and its applications 2006-10, Vol.418 (1), p.11-14
Main Authors: Koliha, J.J., Rakočević, V.
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Linear combinations of projectors
Mathematics
Nullity
Oblique projector
Rank
Sciences and techniques of general use
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2006.01.011