Suppose I have a left-stochastic matrix $\textbf{M}$, an initial probability vector $\textbf{π}$, and a desired probability state, b, which is the ith component of a probability vector $\textbf{b}$. (In other words, I don't care about all of the components of $\textbf{b}$, just one of them.)
I want to find the real power(s) of $\textbf{M}$ that gives me the $\textbf{b}$:
$\textbf{M}^{x}\textbf{π} = \textbf{b} \mid b_{i} = c$
I want to solve for x.
In general, how many solutions could there be? Is there an effective procedure for finding out (1) if a solution exists, (2) the actual solutions?
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Ok, to clarify my question, here's a simple example.
Is there a real number x that makes this equation true?
In general, will there always be such an x?
How can I find that x if it exists?
$\begin{bmatrix} .4 & .6 \\ .6 & .4 \end{bmatrix}^{x}\begin{bmatrix}1 \\ 0\end{bmatrix}=\begin{bmatrix}.5 \\ .5 \end{bmatrix}$