Rescaling positive definite matrices to force a unit eigenvector Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones. 
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$
$X$ and $W$ are all assumed to have real-valued entries, and $X'$ denotes the transpose of $X$.
I don't, yet, have a proof that such a matrix $W$ always exists, but strongly suspect it. Any ideas on algorithms, proofs, or counter-examples would be gratefully received.
The problem arises from work in statistics. 
thanks,
David.
 A: Consider the simplex of nonzero diagonal matrices W with nonnegative entries up to scaling, and the simplex of nonzero vectors V with nonnegative entries up to scaling.
There is a map, $V=\max(WX′XW\mathbf 1,0)$, from the first simplex to the second, with $\max(a,0)$ interpreted entrywise. This is well-defined because $WX'XW\mathbf 1$ always has some positive entry, because the sum of its entries is $1'W X' X W1$, with $W1$ a nonzero vector and $X'X$ positive-definite.
This map clearly sends each k-cell of the first simplex into the corresponding k-cell of the second simplex, since if some of the coordinates of $W$ are $0$ then some of the coordinates of $V$ are $0$.
Every such map on simplices must be surjective. This is because the map from the boundary sphere of one simplex to the boundary sphere of the other is degree one, because every such map on simplices has a boundary-preserving homotopy to the standard isomorphism between those simplices, by induction.
So there is some $W$ such that $\max(WX′XW\mathbf 1,0)=\mathbf 1$. So $WX'XW\mathbf 1=\mathbf 1$.
A: The relevant reference is 
Marshall, A. and Olkin, I. Scaling of Matrices to Achieve Specified Row and Column Sums. Numerische Mathematik  12, 83-90 (1968)
who prove the result in the affirmative for positive definite matrices (and some generalizations). The proof is elegant and construction: the diagonal matrix can be found by minimizing a particular constrained optimization problem. 
There is a good discussion of the problem and its generalizations in
Johnson, C.R. and Reams, R. Scaling of symmetric matrices by positive diagonal congruence. Linear and Multilinear Algebra, 57(2) (2009) 123-140.
-David.
