In math, a Stochastic matrix (also termed probability matrix, transition matrix, substitution matrix or Markov matrix) is a matrix used to describe the transitions of a Markov Chain.
Each of its entries is a nonnegative real number representing a probability.
A right Stochastic matrix is a real square matrix, with row summing to 1.
A Left Stochastic matrix is a real square matrix, with each column summing to 1.
A stationary probability vector $\pi$ is defined as a distribution, that does not change under application of the transition matrix. It is defined as a probability distribution on the set ${1,\dots,n}$ which is also a row eigenvector of the probability matrix, associated with eigenvalue 1.
$\pi P = \pi$