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Please note down all invariant distributions for each of the following transition matrices.

$\begin{pmatrix} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{2}&\frac{1}{2}&0 \end{pmatrix}$, $\begin{pmatrix} 1&0&0&0&\dots\\ 0&1&0&0&\dots\\ 0&0&1&0&\dots\\ 0&0&0&1&\dots\\ \vdots&\vdots&\vdots&\vdots \end{pmatrix}$, $\begin{pmatrix} 1&0&0&0&\dots\\ 1&0&0&0&\dots\\ 0&1&0&0&\dots\\ 0&0&1&0&\dots\\ \vdots&\vdots&\vdots&\vdots \end{pmatrix}$, $\begin{pmatrix} 0&1&0&0&\dots\\ 0&0&1&0&\dots\\ 0&0&0&1&\dots\\ 0&0&0&0&\dots\\ \vdots&\vdots&\vdots&\vdots \end{pmatrix}$

I tried to set up a system of equations for the first matrix. Is the invariant distribution $\begin{pmatrix} \frac{2}{7}\\ \frac{3}{7}\\ \frac{2}{7} \end{pmatrix}$ for the first matrix correct? Are there more possibilities?

I also did it for the second, third and fourth matrix. For each of these three matrices I got the invariant distribution $\begin{pmatrix} \frac{1}{t}\\ \frac{1}{t}\\ \frac{1}{t}\\ \frac{1}{t}\\ \vdots \end{pmatrix}$. Is that correct? Are there more possibilities?

Tino
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  • Please take the time to include key parts of your question as text, using MathJax to format mathematical expressions, instead of pasting pictures of them. Your question is almost incomprehensible without those images, so is inaccessible to people who depend on screen readers. Moreover, images are neither searchable nor show up in summaries. See Formatting and writing from “How to ask a good question.” – amd May 22 '20 at 18:00
  • How are we to interpret all of those ellipses? Are these finite-state systems? What is $t$ in your second invariant distribution? – amd May 22 '20 at 18:05
  • Are you sure that there’s only one stationary distribution for the second matrix? Every vector is an eigenvector of the identity matrix. Similarly for the third: how do you know that there’s only one stationary distribution (or that the one you have is correct)? For example, $(1,0,0,\dots)$ is clearly a stationary distribution for it since the first state is an absorbing state and the rest are all transient. – amd May 22 '20 at 18:06
  • @amd I choose $t$ for the number of entries in every column. Those ellipses demonstrate that there are unlimited entries. I am not sure that there's only one stationary distribution for the second and third matrix. Which other stationary distributions exist? – Tino May 22 '20 at 19:03

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