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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

##### MSC:
 15A21 Canonical forms, reductions, classification
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##### References:
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