Creating a steady state vector

Question

I'm confused on where the intuition came from to divide $w$ by the sum of its entries to find $q$. I don't really see the relation from the sum of its entries with "every solution being a multiple of the solution $w$".

Answer 1

In comments the user JMoravitz gives an answer. The solution to the question says that he wants to find the probability set of all solutions. So he finds a vector that solves the equation: $\vec{w} = (3,4)$ and then you normalize it. So the entries of $\vec{q}$.

Answer 2

This question is quite old, but it's been viewed enough times where I think people are finding it to assist their learning and I feel the other answers aren't providing an explanation beneficial to help learn.

The first stage that you appeared fine with was actually solving out the equation to find a vector $x$ such that $$ Px = x. $$

The first question you had is why are all solutions scalar multiples. I'll show that if $k$ is a scalar then $kx$ is also a solution. Now scalar multiplication commutes and also associates, so we can see that $$ \begin{align} P(kx) = (Pk)x = k(Px) = kx \end{align} $$ since we know that $Px=x$. As such all scalar multiples are also solutions. As for why you cannot have a different vector linearly independent from $x$ as a solution, the null space of $(P-I)$ is one dimensional so if there was another linearly independent vector from $x$, the null space would be two dimensional which isn't possible.

Now to answer your second question of what is the intuition to divide by $w$?, remember that the steady state represents a probability of where you will be after a very very long time. Probabilities have to add up to $1$, so if we pick $$ q = \frac{x}{\sum_{i=1}^2 x_i} $$ we'll end up with a vector of values that has sum $1$.

Why in the book did they choose $w=[3,4]$ instead of something else before dividing? Because integers are pretty and they probably thought it would make the division seem more clear -- which it seems it wasn't haha