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IF we are given a markov chain probability matrix represented by

$$P = \left(\begin{matrix} a&&b&&c\\d&&e&&f\\g&&h&&i\end{matrix}\right)$$

and $a+b+c=d+e+f=g+h+i=1$ with an initial state $(x, y, z)$

Is there a way to know whether after how many iterations the matrix becomes independent of initial state? I know that in excel sheet, we can continue multiplying P a number of times until we get some thing like say

$$P=\left(\begin{matrix} a&&b&&c\\a&&b&&c\\a&&b&&c\end{matrix}\right)$$

but is it possible to find n at which it achieves this state without using an excel sheet ?

Banach Tarski
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SAK
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  • Are you aware of the concept of eigenvalues of a matrix? – Did Sep 22 '16 at 17:01
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    Usually, never. This is a limit case. If, for instance, $P$ has determinant $\neq0$, no power $P^n$ of $P$ may have determinant $0$. – Jean-Claude Arbaut Sep 22 '16 at 17:01
  • Thank you. Does it mean that if determinant of a matrix is 0, then it is possible that for some value of n we will achieve that state? In such cases, is it possible to find n without actually multiply P n times? – SAK Sep 22 '16 at 17:17
  • It never happens that $P\ne P^\infty$ but $P^n=P^\infty$ for some finite $n$. – Did Sep 28 '16 at 13:56

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