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It has been proved that everydoubly substochastic matrixcan be “increased” into a doubly stochastic matrix in the sense that there is a doubly stochastic matrixwith([Marshall AW, Olkin I, Arnold BC. Inequalities; theory of majorization and its applications. 2nd ed. New York (NY): Springer; 2011], Theorem C.1). In this paper, we show this assertion does not necessarily hold for all infinite doubly substochastic matrices. Then we characterize all such matrices which are called  increasable doubly substochastic matrices.  We also provide an application of this result to the existence of a non-negative solution for some infinite linear equations.

Increasable doubly substochastic matrices with application to infinite linear equations