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Nonsingularity of the difference and the sum of two idempotent matrices. (English) Zbl 1195.15005

It is shown that for any two idempotent matrices \(P, Q\) the nonsingularity of \(P-Q\) is equivalent to the nonsingularity of any combination \(aP+bQ-cPQ\) (\(a, b\neq 0, \;a+b=c\)), and the nonsingularity of \(P+Q\) is equivalent to the nonsingularity of any combination \(aP+bQ-cPQ\) (\(a, b\neq 0,\;a+b\neq c\)).

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A03 Vector spaces, linear dependence, rank, lineability
15A24 Matrix equations and identities
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