TITLE: Rescaling positive definite matrices to force a unit eigenvector
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: 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.