Why Markov matrices always have 1 as an eigenvalue

Solution 1:

I just encounter the same problem as you do, here is my solution.

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Solution 2:

My intuition is that since $A$ describes a transition from some vector that encodes probability distribution to another vector that also encodes probability distribution, $A$ is not allowed to scale up or scale down the vector along the same direction, because otherwise that vector will no longer have all its entries add up to $1$ and it would no longer be a probability distribution.

In other words, suppose $A$ is a markov matrix, and $v$ is a vector that represents some probability distribution. This means that all the entries in $v$ have to add up to one.

Now let $v'= vA$. If $v$ were an eigenvector, that means that $v' = \lambda v$. Since $v'$ is also a probability distribution, all its entries must also add up to $1$, and this can only happen if $\lambda = 1$.