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.


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


Zbl 0946.15020
Full Text: DOI


[1] Buckholtz, D, Inverting the difference of Hilbert space projections, Amer. math. monthly, 104, 60-61, (1997) · Zbl 0901.46019
[2] Buckholtz, D, Hilbert space idempotents and involutions, Proc. amer. math. soc., 128, 1415-1418, (2000) · Zbl 0955.46015
[3] Groß, J; Trenkler, G, Nonsingularity of the difference of two oblique projectors, SIAM J. matrix anal. appl., 21, 390-395, (1999) · Zbl 0946.15020
[4] Koliha, J.J, Range projections of idempotents in \(C\^{}\{∗\}\)-algebras, Demonstratio math., 34, 91-103, (2001) · Zbl 0981.46047
[5] Koliha, J.J; Rakočević, V, On the norms of idempotents in \(C\^{}\{∗\}\)-algebras, Rocky mountain J. math., 34, 685-697, (2004) · Zbl 1066.46044
[6] Ljance, V.E, Some properties of idempotent operators (in Russian), Teor. i prikl. mat. L’vov, 1, 16-22, (1959)
[7] Marsaglia, G; Styan, G.P.H, Equalities and inequalities for the rank of matrices, Linear multilinear algebra, 2, 269-292, (1974)
[8] Pták, V, Extremal operators and projectors, Časopis Pěst. mat., 110, 343-350, (1985) · Zbl 0611.47022
[9] Rakočević, V, On the norm of idempotent in a Hilbert space, Amer. math. monthly, 107, 748-750, (2000) · Zbl 0993.47009
[10] Tian, Y; Styan, G.P.H, Rank equalities for idempotent and involutory matrices, Linear algebra appl., 335, 101-117, (2001) · Zbl 0988.15002
[11] Vidav, I, On idempotent operators in a Hilbert space, Publ. inst. math. (beograd), 4, 18, 157-163, (1964) · Zbl 0125.35101
[12] Wimmer, H.K, Canonical angles of unitary spaces and perturbations of direct complements, Linear algebra appl., 287, 373-379, (1999) · Zbl 0937.15002
[13] Wimmer, H.K, Lipschitz continuity of projectors, Proc. amer. math. soc., 128, 873-876, (2000) · Zbl 0935.46014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.