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When is a matrix a sum of idempotents? (English) Zbl 0696.15011
Let M be an $$n\times n$$ matrix with elements in an algebraically closed field. The authors prove that M is a sum of idempotents if and only if $$tr(M)=ke$$ where k is an integer with $$k\geq rank M$$. They also prove that if A is an $$n\times n$$ matrix of integers then A is a sum of integral idempotents if rank A$$<tr(A)$$, and is not a sum of integral idempotents if rank A$$>tr(A)$$. If rank A$$=tr(A)$$, A may or may not be a sum of integral idempotents.
Reviewer: F.J.Gaines

##### MSC:
 15A21 Canonical forms, reductions, classification 15B36 Matrices of integers
##### Keywords:
trace; rank; integer matrix; idempotents
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##### References:
 [1] DOI: 10.1017/S0017089500000173 · Zbl 0157.07101 [2] Hartwig R. E., Math. Japonica 25 pp 1– (1980) [3] Hartwig R. E., Lin. Multilin. Alg. 25 (1980) [4] Hoffman, K. 1961.Linear Algebra, 213–213. NJ: Prentice-Hall. Englewood Cliffs. [5] Newman M., Lin. Multilin. Alg. 16 pp 339– (1984) [6] Newman M., Integer Matrices (1972)
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