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

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.

### MSC:

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

### Keywords:

projector; direct sum; inverse; non-singularity

Zbl 0946.15020
Full Text:

### References:

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