The ideas are simple. Instead of having a double sum, we can have a matrix product of the form 1_m^T X 1_n, which is nothing more than a compact way of writing

The ideas are simple. Instead of having a double sum, we can have a matrix product of the form 1_m^T X 1_n, which is nothing more than a compact way of writing the sum of all the entries of matrix X. Note that vec(X) is a mn-dimensional vector that contains the n stacked columns of X. Hence, 1_m^T X 1_n = 1_mn^T vec(X) which is an inner product. Note that the equality 1_mn^T vec(X) = c defines a hyper-plane, whereas the inequality 1_mn^T vec(X) <= c defines a half-space: http://en.wikipedia.org/wiki/Half-space http://en.wikipedia.org/wiki/Half-space The intersection of half-spaces defines a polytope, which is nothing more than the higher dimensional equivalent of a polyhedron. Take a look at Boyd & Vanderberghe's book on Convex Optimization for details: http://www.stanford.edu/~boyd/cvxbook/ http://www.stanford.edu/~boyd/cvxbook/