Properties of the transition matrix:

We show that the matrix $(I-\lambda P_d)^{-1}$ is order conserving.

Lemma 5.3   The following holds for a probability matrix P and $0 \leq\lambda<1$:

1.

If $\Vert\vec{u}\Vert \ \geq \ 0$ then $\Vert(I-\lambda P)^{-1}\vec{u}\Vert \ \geq \ \Vert\vec{u}\Vert \ \geq \ 0$

2.

If $\Vert\vec{u}\Vert \ \geq \ \Vert\vec{v}\Vert$ then $\Vert(I-\lambda P)^{-1}\vec{u}\Vert \ \geq \ \Vert(I-\lambda P)^{-1}\vec{v}\Vert$

3.

If $\Vert\vec{u}\Vert \ \geq \ 0$ then $\Vert\vec{u}^{T}(I-\lambda P)^{-1}\Vert\ \geq\ \Vert\vec{u}^{T}\Vert \geq 0$

Proof:Since $\Vert P\Vert= 1$ then $\Vert\lambda P\Vert\leq 1$. By theorem

$$\begin{eqnarray*}(I - \lambda P_{d})^{-1}\vec{u} = \vec{u} + \underbrace{(\lambd... ... + \ldots}_{(sum\ of\ positive\ vectors )} \geq \vec{u} \geq 0 \end{eqnarray*}$$

$\Box$