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Nonsingularity of linear combinations of idempotent matrices. (English) Zbl 1081.15017

The authors show that, for idempotent \(n\times n\) complex matrices \(P_1\) and \(P_2\), the nonsingularity of \(P_1+P_2\) is equivalent to the nonsingularity of \(c_1P_1+c_2P_2\), where \(c_1\) and \(c_2\) are nonzero complex numbers satisfying \(c_1+c_2\neq0\).

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
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References:

[1] Groß, J; Trenkler, G, Nonsingularity of the difference of two oblique projectors, SIAM J. matrix anal. appl., 21, 390-395, (1999) · Zbl 0946.15020
[2] J.J. Koliha, V. Rakočević, I. Straškraba, The difference and sum of projectors, Linear Algebra Appl., in press (doi:10.1016/j.laa.2004.03.008)
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