Editing Magic square (section)

===Birkhoff–von Neumann decomposition===
Dividing each number of the magic square by the magic constant will yield a [[doubly stochastic matrix]], whose row sums and column sums equal to unity. However, unlike the doubly stochastic matrix, the diagonal sums of such matrices will also equal to unity. Thus, such matrices constitute a subset of doubly stochastic matrix. The Birkhoff–von Neumann theorem states that for any doubly stochastic matrix <math>A</math>, there exists real numbers <math>\theta_1,\ldots,\theta_k \ge 0</math>, where <math>\sum_{i=1}^k \theta_i = 1</math> and [[permutation matrices]] <math>P_1,\ldots,P_k</math> such that

:<math>A = \theta_1 P_1 + \cdots + \theta_k P_k. </math>

This representation may not be unique in general. By Marcus-Ree theorem, however, there need not be more than <math> k \le n^2 - 2n + 2</math> terms in any decomposition.<ref>{{cite journal|last1=Marcus|first1=M.|last2=Ree|first2=R.|title=Diagonals of doubly stochastic matrices|journal=The Quarterly Journal of Mathematics|date=1959|volume=10|issue=1|pages=296–302|doi=10.1093/qmath/10.1.296}}</ref> Clearly, this decomposition carries over to magic squares as well, since a magic square can be recovered from a doubly stochastic matrix by multiplying it by the magic constant.