The difference and sum of projectors
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 \textit{J. Gross } and \textit{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.
scientific article; zbMATH DE number 2105747