# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] DOI: 10.1093/qmath/7.1.76 · Zbl 0071.01501 [2] Ben Israel A., Generalized Inverses: Theory and Applications (1974) [3] Flanders H., Proc. Amer. Math. Soc. 2 pp 871– (1951) [4] Gantmacher F. R., The Theory of Matrices 1 (1960) · Zbl 0088.25103 [5] Mäkeläinen T., Sankhya 38 pp 400– (1976) [6] Jacobson, N. 1953.Lectures in Linear Algebra, 4–4. New York: Van Nostrand. · Zbl 0053.21204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.