## The nullity and rank of linear combinations of idempotent matrices.(English)Zbl 1104.15001

J. K. Baksalary and O. M. Baksalary [Linear Algebra Appl. 388, 25–29 (2004; Zbl 1081.15017)] have shown that if $$P_1$$, $$P_2$$ are idempotent matrices (i.e. $$P_j^2=P_j$$), then the nonsingularity of $$P_1+P_2$$ is equivalent to the one of any linear combination $$P:=c_1P_1+c_2P_2$$, $$c_j\in {\mathbb C}^*$$, $$c_1+c_2\neq 0$$. In the present note the authors show that the nullity (i.e. the dimension of the nullspace) and rank of $$P$$ are constant. They provide a simple proof of a rank formula from J. Groß and G. Trenkler [SIAM J. Matrix Anal. Appl. 21, 390–395 (1999; Zbl 0946.15020)].

### MSC:

 15A03 Vector spaces, linear dependence, rank, lineability 15A24 Matrix equations and identities

### Citations:

Zbl 1081.15017; Zbl 0946.15020
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### References:

 [1] Baksalary, J.K.; Baksalary, O.M., Nonsingularity of linear combinations of idempotent matrices, Linear algebra appl., 388, 25-29, (2004) · Zbl 1081.15017 [2] Groß, J.; Trenkler, G., Nonsingularity of the difference of two oblique projectors, SIAM J. matrix anal. appl., 21, 390-395, (1999) · Zbl 0946.15020 [3] Koliha, J.J.; Rakočević, V.; Straškraba, I., The difference and sum of projectors, Linear algebra appl., 388, 279-288, (2004) · Zbl 1060.15011 [4] Koliha, J.J.; Rakočević, V., Fredholm properties of the difference of orthogonal projections in a Hilbert space, Integral equations operator theory, 52, 125-134, (2005) · Zbl 1082.47009 [5] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear and multilinear algebra, 2, 269-292, (1974)
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