How can I make sense of this Matrix inequality?

Question

In https://tel.archives-ouvertes.fr/tel-02926037/document one page 35 the following Lemma is stated:

Lemma I.5: Let $M \in (\Bbb R^+)^{N \times N}$ be a stochastic matrix. Then for any positive vector $x\in (\Bbb R_*^+)^N,$ one has $$ \frac 1{Mx} \leq M \frac 1x $$

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where the division is to be interpreted component wise. How can I make sense or even proof this inequality?

Answer 1

The meaning seems to be pretty clear: $1\leq \left(\sum_j M_{i,j}x_j\right)\left(\sum_j M_{i,j}/x_j\right)$ for every $i=1,\dots,N$.

This follows from Cauchy-Schwarz: $$ 1 = \sum_{j=1}^N M_{i,j} = \sum_{j=1}^N \sqrt{x_j} \frac1{\sqrt{x_j}} M_{i,j} \leq \left(\sum_{j=1}^N x_j M_{i,j}\right)^{1/2}\left(\sum_{j=1}^N \frac1{x_j} M_{i,j}\right)^{1/2} . $$