Define:
- Doubly Stochastic Matrix is a square matrix of non-negative real numbers, each of whose rows and columns sums to 1.
- Probability vector is a vector of non-negative real numbers whose sum is 1.
- Permutation Matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere
- Convex Combination is a linear combination whose coefficients sum to 1
I want to show that for every Doubly Stochastic Matrix M there are finitely many permutation matrices and coefficients such that M can be written as a convex combination of those permutation matrices using those coefficients.
I know I'm supposed to use that fact that there exists $\pi$ a probability vector such that $M\pi=\pi$, and I managed to show that. I also know I'm supposed to use the fact that every point in a polytope can be written as a convex combination of that polytope vertices.
I thought about proving it by induction on n (where M is an $n\times n$). The base is trivial. For the step we have M a Doubly Stochastic matrix of order $n\times n$.
I can take a vertex in the polytope $P_n$ defined by
\begin{array}{rcl}
\left(1\right) & \underset{j\in\left[n\right]}{\sum}M_{i,j}=1 & ,\forall i\in\left[n\right]\\
\left(2\right) & \underset{i\in\left[n\right]}{\sum}M_{i,j}=1 & ,\forall j\in\left[n\right]\\
\left(3\right) & M_{i,j}\geq0 & ,\forall i,j\in\left[n\right]
\end{array}
The (1) and (2) equations are linearly dependent, and we can leave one out. So we have $n^2+2n-1$ inequalities. We know that for every vertex $X$ in $P_n$, $n^2$ of the inequalities are actually equal for X, that is, there are at least $n^2-2n-1$ $X_{i,j}=0$.
Since X is Doubly Stochastic with $n^2-2n-1$ zeros, there is a line with n-1 zeros. We can show that since the average of zeroes for each row is $\frac{n^2-2n-1}{n}>n-2$ by the Pigeonhole principle there exists one with at least n-1 zeros (and since its stochastics, its exactly n-1).
So now we can remove that row from X, and remove the column which contains 1 in that row, and get X' a Doubly Stochastic matrix of order n-1, apply the induction hypothesis, and get a finite number of convex combinations of permutation matrices that equals X'. From that we can construct a finite number of convex combinations for X.
This is where I'm stuck:
- I'm not sure how to show these are the only combinations possible.
- Assuming I did show 1., then I know M can be written as a convex combination of vertices of $P_n$ (such as X that we worked on). But how do I know there is a finite number of such convex combinations?
- I'm not using the fact mentioned before, that there exists exists $\pi$ a probability vector such that $M\pi=\pi$, which makes me think I'm missing the point.
Would appreciate some help,
Thanks!
