<s>[INST] Let G be a doubly stochastic matrix, that is, a square matrix whose entries are all nonnegative and whose rows and columns sum to 1. Let v be any n x 1 column vector of probabilities. Prove that the matrix A, defined as G IID kostenG, is doubly stochastic.

I want to prove that the matrix G has the following properties:
1) A has nonnegative entries
2) the sum of the rows is 1, and
3) the sum of the columns is 1.

Proof
Because entries of G are non-negative, entries of A will be nonnegative as well. The scalar product A = kv with k being scalar 1 is, A = G".V..Now the rows of the resulting matrix A simply represents the dot product of row vectors of matrix G and column vector V representing the following:
AV(G\({}^{\text{T}}\)V)
= \(\text{\Vert(\text{x}_{1}{}^{\text{T}}\text{x}_{1})+\text{...}\ (\text{x}_{2}{}^{\text{T}}\text{x}_{2})+...\ (\text{x}_{\text{m}}{}^{\text{T}}\text{x}_{\text{m}})^{\text{T}}}\)where \(x_{i}\)'s are the column vectors of \(G\) and the transpose of v.
Because v is a probability vector, entries of v (or v\({}^{T}\)) are all positive. Thus, we have:
1) A has nonnegative entries.
2) The sum of the entries of a row is just the dot product of that row and host, the vector of all ones. That is precisely the dot product of the transpose of that row (EBVR implementation of a point in the hyperpolate) and representable of all ones in the space, thus counting equal 1.
At this point, I am stuck
:
Lucky. Any help?[/INST]