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)].


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