On the existence of sequences and matrices with prescribed partial sums of elements

We prove necessary and sufficient conditions for the existence of sequences and matrices with elements in given intervals and with prescribed lower and upper bounds on the element sums corresponding to the sets of an orthogonal pair of partitions. We use these conditions to generalize known results on the existence of nonnegative matrices with a given zero pattern and prescribed row and column sums. We also generalize recently proven results on the existence of (a real or nonnegative) square matrix A with a given zero pattern and with prescribed row sums such that A + AT is prescribed. We also introduce Hadamard adjustments, by means of which we generalize known results on the scaling of matrices with a given pattern to achieve prescribed row and column sums.