The nearest 'doubly stochastic' matrix to a real matrix with the same first moment

Let T be an arbitrary n x n matrix with real entries. We consider the set o f all matrices with a given complex number as an eigenvalue, as well as bei ng given the corresponding left and right eigenvectors. We find the closest matrix A, in Frobenius norm, in this set to the matrix T. The normal cone to a matrix in this set is also obtained. We then investigate the problem o f determining the closest 'doubly stochastic' (i.e., Ae = e and e(T)A = e(T ), but not necessarily non-negative) matrix A to T, subject to the constrai nts e(1)(T)A(k)e(1) = e(1)(T)T(k)e(1),for k = 1, 2,... A complete solution is obtained via alternating projections on convex sets for the case k = 1, including when the matrix is non-negative. (C) 1998 John Wiley & Sons, Ltd.