I am just trying to consider the classical discrete-time Markov Chain problem. Consider the transition matrix P, which transforms state vector $x(k)$ to $x(k+1)$, satisfying:
$x(k+1)$ = $P*x(k)$
It is well known that the principal eigenvalue of P is $1$ and its corresponding right eigenvector is all 1 while its left eigenvector determines the expectation arrival rate of every state.
My question is that whether I can adjust the direction of the associated principal left eigenvector so I can control the final allocation ratio of every state.
The kind of adjustification can be made by re-allocating weights of elements in every row of P. I am trying to find out the limit where I can control the direction of the expected eigenvector. And if possible, I want to find a definite way to drive the eigenvector to the designated direction.
Any suggestion is warmly welcomed.
If your desired left eigenvector is a row vector $x$ with $x_i\ge0$ and $\sum x_i =1$, then the matrix $$ P= \begin{pmatrix} x \\ x\\ \vdots \\ x\end{pmatrix} $$ has $x$ as left eigenvector to the eigenvalue $1$.