Given a data matrix $D\in\mathbb{R}^{N \times N}$

Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$,

if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j (Q^B_{ij}D_{ij})]^2$ then $\sum_i\sum_j (Q^A_{ij}M_{ij}) \geq \sum_i\sum_j (Q^B_{ij}M_{ij})$?

p.s. Sorry I don't know how to name the problem so the title is just "Question about inearization". If you have any good idea please tell me.

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Do you have a simple interpretation of the condition $\big(\sum_i\sum_j (Q^A_{ij}D_{ij})\big)^2 \geq \big(\sum_i\sum_j (Q^B_{ij}D_{ij})\big)^2$? That would (i) make it easier for others to understand the problem, and (ii) help you choose a better title. :) –  Rahul Jan 25 '13 at 6:53
@RahulNarain I am not sure but maybe it can be represented in terms of Hadamard product $||Q^A\circ D||^2_F \geq ||Q^B\circ D||^2_F$? –  Rein Jan 26 '13 at 5:49
I don't know it will be useful but but another formulation for the sums is $\sum_i\sum_j(Q_{ij}^AD_{ij})=Tr[Q^{A\,T}D]$, where the $T$ denotes the transpose and $Tr$ the trace of a matrix. –  Sebastien B Jan 27 '13 at 13:09
@SebastienB Thanks for the suggestion. –  Rein Jan 28 '13 at 11:50