Understanding the basics

We can always expand $\boldsymbol{w}(0)$ in terms of the right eigenvectors $\boldsymbol{v}$ of $\boldsymbol{W}$ as

$$\begin{equation*} \boldsymbol{w}(0) = \sum_i\alpha_i\boldsymbol{v}_i, \end{equation*}$$

resulting in

$$\begin{equation*} \boldsymbol{w}(t) = \boldsymbol{W}^t\boldsymbol{w}(0)=\boldsymbol{W}^t\sum_i\alpha_i\boldsymbol{v}_i= \sum_i\lambda_i^t\alpha_i\boldsymbol{v}_i, \end{equation*}$$

with $\lambda_i$ the $i^{\mathrm{th}}$ eigenvalue corresponding to the eigenvector $\boldsymbol{v}_i$.

If we assume that $\lambda_0$ is the largest eigenvector we see that in the limit $t\rightarrow \infty$, $\boldsymbol{w}(t)$ becomes proportional to the corresponding eigenvector $\boldsymbol{v}_0$. This is our steady state or final distribution.

• «
• 1
• ...
• 23
• 24
• 25
• 26
• 27
• 28
• 29
• 30
• 31
• 32
• 33
• 34
• 35
• 36
• 37
• 38
• 39
• 40
• ...
• 50
• »