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On linear combinations of two idempotent matrices over an arbitrary field. (English) Zbl 1206.15014
Given two non-zero scalars $$\alpha$$ and $$\beta$$ in an arbitrary field, the author gives necessary and sufficient conditions for a square matrix over the field to be a linear combination of two idempotents, with the prescribed coefficients $$\alpha$$ and $$\beta$$. The conditions are in terms of the Jordan canonical form of the matrix. The form of the conditions depends on whether $$\alpha=\pm\beta$$ and on whether the ground field is of characteristic 2. The case $$\alpha=1$$, $$\beta=-1$$, over an algebraically closed field of characteristic zero, was already worked out by R. E. Hartwig and M. S. Putcha [Linear Multilinear Algebra 26, No. 4, 267–277 (1990; Zbl 0696.15010)].

MSC:
 15A24 Matrix equations and identities 15A23 Factorization of matrices
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References:
 [1] Flanders, H., Elementary divisors of AB and BA, Proc. amer. math. soc., 2, 871-874, (1951) · Zbl 0044.00602 [2] Gantmacher, F.R., The theory of matrices, Vol. 1, (1960), Chelsea New York · Zbl 0088.25103 [3] Hartwig, R.E.; Putcha, M.S., When is a matrix a difference of two idempotents?, Linear and multilinear algebra, 26, 267-277, (1990) · Zbl 0696.15010
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