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The nullity and rank of linear combinations of idempotent matrices. (English) Zbl 1104.15001
{\it J. K. Baksalary} and {\it 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 $P:=c_1P_1+c_2P_2$, $c_j\in {\Bbb C}^*$, $c_1+c_2\neq 0$. 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 {\it J. Groß} and {\it G. Trenkler} [SIAM J. Matrix Anal. Appl. 21, 390--395 (1999; Zbl 0946.15020)].

##### MSC:
 15A03 Vector spaces, linear dependence, rank 15A24 Matrix equations and identities
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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) [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) · Zbl 0297.15003