Choi, Man-Duen; Wu, Pei Yuan Convex combinations of projections. (English) Zbl 0721.15007 Linear Algebra Appl. 136, 25-42 (1990). The paper contains the following results: On an n-dimensional inner- product space, every operator T that satisfies \(0\leq T\leq 1\) is a convex combination of as few as \([\log_ 2n]+2\) projections and this number is sharp. If \(0\leq T\leq 1\) and trace T is a rational number, then T is an average of projections. Further results are also got for the cases when the projections are needed to have the same rank / or to be commuting. In each case, the optimal number of projections is found out. The convex combination part presented in this paper is very interesting. Reviewer: S.Sridhar (Madras) Cited in 12 Documents MSC: 15A30 Algebraic systems of matrices 15A63 Quadratic and bilinear forms, inner products 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) Keywords:convex combinations of projections; inner-product space; average of projections; optimal number of projections PDFBibTeX XMLCite \textit{M.-D. Choi} and \textit{P. Y. Wu}, Linear Algebra Appl. 136, 25--42 (1990; Zbl 0721.15007) Full Text: DOI References: [1] Davis, C., Separation of two linear subspaces, Acta Sci. Math., 19, 172-187 (1958) · Zbl 0090.32603 [2] Fillmore, P. A., On sums of projections, J. Funct. Anal., 4, 146-152 (1969) · Zbl 0176.43404 [3] Fong, C. K.; Murphy, G. J., Averages of projections, J. Operator Theory, 13, 219-225 (1985) · Zbl 0614.47011 [4] Horn, R. A.; Johnson, C. R., Matrrix Analysis (1985), Cambridge U.P.,: Cambridge U.P., Cambridge [5] Kadison, R. V.; Pedersen, G. K., Means and convex combinations of unitary operators, Math. Scand., 57, 249-266 (1985) · Zbl 0573.46034 [6] Nakamura, Y., Any Hermitian matrix is a linear combination of four projections, Linear Algebra Appl., 61, 133-139 (1984) · Zbl 0547.15012 [7] Nishio, K., The structure of a real linear combination of two projections, Linear Algebra Appl., 66, 169-176 (1985) · Zbl 0584.47014 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.