Theorem 1: The product of two stochastic matrixes is a stochastic matrix.
It is clear that in the above representation, matrix A is a stochastic matrix.
Matrix M is a stochastic matrix since all the elements in a row of M sum to one and each element lies in the range [0, 1].
Therefore, C is a matrix with nonnegative entries and row-sums of equal to 1, and thus, it is a stochastic matrix.
Since a transition matrix is a stochastic matrix, i.e., the values of the cells in each row must add up to one, it has an eigenvalue equal to one while its other eigenvalues are smaller or equal to one.
Since B is a stochastic matrix and x(t + 1) = B(x t),t).x t), we have min j x j t ≤ x i t + 1 ≤ max j x j t, i = 1, …, N, (22).
Theorem 2 (Convergence Theorem): Considering opinion update rule (2), suppose we have a matrix B t1,t2) = [b ij (t1,t2)], which is a stochastic matrix and models accumulated weights where b ij is an element of matrix B the sequences 0 = t0 < t1 < t2 < … ≤ T and δ1, δ2, …, δ i, … are such that 0 ≤ δ t ≤ 1 and ∑ t = 0 ∞ δ t = ∞.
There are no negative summands involved, as P is a stochastic matrix and λ > 0.
∏ being a stochastic matrix, ∏ n -1 is also stochastic, it is therefore clear that the sum of R n -1 over the columns gives 1 - y n -2 and the corollary is proved.
