Equilibrants, semipositive matrices, calculation and scaling

For square, semipositive matrices A (Ax>0 for some x>0), two (nonnegative) equilibrants e(A) and E(A) are defined. Our primary goal is to develop theory from which each may be calculated. To this end, the collection of semipositive matrices is partitioned into three subclasses for each equilibrant, and a connection to those matrices that are scalable to doubly stochastic matrices is made. In the process a certain matrix/vector equation that is related to scalability of a matrix to one with line sums 1 is derived and discussed.