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SUM OF PRODUCTS


This quiz is taken from the book "Statistical Mechanics: Rigorous Results" by D. Ruelle *Addison-Wesley 1969).


Let Aij be a nxn symmetric matrix, with |Aij|<1. Consider the following sum,

P(k,n) = ∑S (-1)|S*{1,..,k}|i in Sj in S' Aij

where:


For exaample, consider the case n=2:

Since A21=A12, P(1,2)=0, and since |A12|<1, P(0,2) and P(2,2) are positive.


Show that P(k,n) is greater or equal to 0 for every n and k (between 0 and n).

Marco Corvi - 2006