# Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector.

What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$?

In addition if $\Vert x \Vert_2=1$, what can I say about $\Vert Dx \Vert_2$?

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If $\|x\|_2$...what? You , miss something in your second question. –  1015 Apr 10 '13 at 3:13
@julien Thanks, updated. –  Rein Apr 10 '13 at 3:42
From here, we have that the largest eigenvalue of a stochastic matrix is $1$. Hence, we have $$\Vert Dx \Vert_2 \leq \Vert x \Vert_2$$