# There exists a doubly stochastic matrix which maps a vector into any other vector with the same average value

Let $x\in \mathbb{R}^n$ be any vector. Let $y \in \mathbb{R}^n$ be any vector such such that $\sum_{i=1}^nx(i)=\sum_{i=1}^n y (i)$. Show or disprove that there exists a doubly stochastic matrix $M(y) \in \mathbb{R}^{n\times n}$ such that $x=M(y)y$.

• This looks like homework, and so it should probably have the associated tag. However, a hint: what if $y = (1,1,\ldots,1)$? Jun 21 '18 at 16:22
• It's actually just a simple question I had in mind and wanted to check. By the way, I edited the question, I think it's clearer now. I don't want to find $M$ which holds for any $y$, I want to show that, given any $y$ with the same average, there is some $M$ such that $x=My$ for that specific $y$. Jun 21 '18 at 16:35

Let $n=2$, $y=(1,1)^{\intercal}$, and $x=(0,2)^{\intercal}$. Suppose that there exists a right stochastic matrix $M=(m_{ij})$ satisfying $My=x$. Then, $[My]_1=x_1$.
We know $[My]_1 = m_{11} + m_{12}$ and $x_1=0$. What can you conclude from this?