SUM DECOMPOSITIONS OF SYMMETRICAL MATRICES

Given a symmetric n X n matrix A and n numbers r1,..., r(n), necessary and sufficient conditions for the existence of a matrix B, with a giv en zero pattern, with row sums r1,..., r(n), and such that A = B + B(T ) are proven. If the pattern restriction is relaxed, then such a matri x B exists if and only if the sum r1 + ... + r(n) is equal to half the sum of the elements of A. The case where A and B are nonnegative matr ices is solved as well.