A (usually square) matrix for which the sum of each row and each column equals one. An example is a transilient matrix used to describe vertically nonlocal turbulent mixing in the atmosphere, where the sum of each column must equal one to conserve a state such as heat or moisture, and the sum of each row must equal one to conserve air mass. The largest eigenvector of a doubly stochastic matrix is not larger than unity, which implies absolute numerical stability for any time step and any grid spacing when using the matrix to make forecasts of turbulent mixing.