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When is a matrix a difference of two idempotents? (English) Zbl 0696.15010
Let A be an \(n\times n\) matrix with elements in a field of characteristic zero. The authors prove that \(A=E-F\), where E and F are idempotents if, and only if, A is similar to a block diagonal matrix diag(N,D,-D,I-XY,YX- I), where N is nilpotent, D has no zero or \(\pm 1\) eigenvalues, and XY and YX are nilpotent. They also prove that a matrix A is idempotent if and only if rank(I-A)\(\leq tr(I-A)\), the eigenvalues of A are nonnegative and the group inverse \(A^{\#}\) exists.
(In the statement of Lemma 2 the condition \(\sigma (P)\cap \sigma (Q)=\emptyset\) should read \(\sigma (-P)\cap \sigma (Q)=\emptyset)\).
Reviewer: F.J.Gaines

15A21 Canonical forms, reductions, classification
Full Text: DOI
[1] DOI: 10.1093/qmath/7.1.76 · Zbl 0071.01501
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