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        riddles >> putnam exam (pure math) >> Decomposing Projection Operators
        
(Message started by: JocK on Jan 28th, 2006, 6:25am)
Title: Decomposing Projection Operators
Post by JocK on Jan 28th, 2006, 6:25am
Consider two non-commuting projection operators P and Q on a finite-dimensional inner-product space (i.e. two operators which are idempotent P2 = P, Q2 = Q and for which PQ =/= QP). Prove that it is not possible to construct non-negative operators R00, R01, R10, R11 (i.e. Hermitean operator with non-negative eigenvalues) such that:

R00 + R01 = P

R10 + R11 = I - P

R00 + R10 = Q

R01 + R11 = I - Q

(I is the identity operator).


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