The difference and sum of projectors. (English) Zbl 1060.15011

Koliha, J. J.; Rakočevič, V.; Straškraba, I.

Linear Algebra Appl. 388, 279-288 (2004).

The authors give simple proofs, without reference to rank theory for matrices, of some results on the non-singularity of the difference \(P-Q\) of projections \(P\) and \(Q\) obtained by J. Gross and G. Trenkler [SIAM J. Matrix Anal. Appl. 21, No. 2, 390–395 (1999; Zbl 0946.15020)]. In the proofs, the relations among the ranges and the null-spaces of projections \(P\) and \(Q\) have been employed. A new characterization of the non-singularity of \(P-Q\) in terms of the non-singularity of \(P+Q\) has been obtained. The authors also give necessary and sufficient conditions for the non-singularity of \(P+Q\) and explicit formulae for the inverse of \(P+Q\) separately in the case when \(P-Q\) is non-singular, and when \(P-Q\) is singular.

Reviewer: Hong-Ke Du (Xi’an)

MSC:

15A09	Theory of matrix inversion and generalized inverses
15A24	Matrix equations and identities

Keywords:

projector; direct sum; inverse; non-singularity

Citations:

Zbl 0946.15020

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