In the comments on EDP18 we are considering a certain decomposition problem that can be understood in its own right. At various points I have asserted that if we can find a decomposition of a particular kind then we will have a positive solution to EDP. And at various points in the past I have even sketched proofs of this. But I think it would be a good idea to do more than merely sketch a proof. So in this post I shall (I hope) give a completely rigorous derivation of EDP from the existence of an appropriate decomposition. (Well, I may be slightly sketchy in places, but only about details where it is obvious that they can be made precise.) I shall also review some material from earlier posts and comments, rather than giving links.

**Representing diagonal matrices**

First, let me briefly look again at how the ROD (representation of diagonal) approach works. If and are HAPs, I shall write for the matrix such that if and 0 otherwise. The main thing we need to know about is that for every

Suppose now that is a diagonal matrix with diagonal entries and that we can write it as where each and each is a HAP. Then

If and for every then it follows that there exists such that

and from that it follows that there is a HAP such that So if we can make arbitrarily small, then EDP is proved.

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