Deterministic stochastic matrices (DSM) consist of 0 and 1 only. Decomposition of a stochastic matrix (SM) means representation of it as a linear combination of DSMs with positive coefficients. In the classical (non-constructive) setting, any m x n SM can be represented as a linear combination of maximum m(n–1)+1 DSMs. In the constructive setting, SM consists of constructive real numbers, and the decomposition must be produced by an algorithm. It is proved in the article that, in the constructive setting, any m x n SM can be represented as a linear combination of maximum m(n–1)+min(m, n) DSMs. This estimate is known to be exact only for m=1 and m=2. It is conjectured that the exact estimate could be m(n-1)+2 for all m, n.
Original title: К.М.Подниек. О конструктивном разложении стохастических матриц. Автоматика и вычислительная техника, 1972, 3, 18-20.
