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So the big theory is every hilbert space operator is a linear combination of projections, and for matrices, Any Hermitian Matrix is the Linear Combination of Four Projections.

But the proof in the paper does not give a explicit algorithm to compute those four projections, thus I am curious how we should look for those four projections.

Thanks!

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    @HuiYu: This is a consequence of the spectral theorem for normal matrices: http://en.wikipedia.org/wiki/Spectral_theorem#Normal_matrices. You can check directly that a linear combination of commuting self-adjoint idempotents is normal. Conversely, the spectral theorem says that a normal operator has an orthogonal basis of eigenvectors, so the projections in question can be taken to be the othogonal projections onto the eigenspaces. (This has probably zero bearing on your question.)2012-07-08

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V I Rabanovich, On the decomposition of a diagonal operator into a linear combination of idempotents or projections, Ukrainian Math. J. 57 (2005), no. 3, 466–473, MR2188435 (2006g:47001), writes, "We prove (Theorem 2) that a diagonal self-adjoint operator is a linear combination of four orthoprojectors." I haven't tried to work through the details, but I think there may be enough there to answer your question.