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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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