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The question gives a $2\times2$ transition matrix: $$ \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix}. $$

And then it gives me the initial state matrix but I'm wondering how do I find the initial state matrix by myself?

The initial state matrix of this transition matrix is $$ \begin{bmatrix} 0.4 & 0.6\\ \end{bmatrix}. $$

user151764
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1 Answers1

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If you are given the final state matrix, then you could apply the inverse of the transition matrix to the final state matrix to get the initial state matrix. $$ \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix}^{-1} =\begin{bmatrix} 1.4 & -0.4\\ -0.6 & 1.6 \end{bmatrix} $$

Normally, you are given the initial state matrix and you apply the transition matrix to get the final state matrix (which may become the initial state matrix for the next transition). For the given initial state matrix, the final state matrix would be $$ \begin{bmatrix} 0.4 & 0.6\\ \end{bmatrix} \begin{bmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{bmatrix} = \begin{bmatrix} 0.5 & 0.5\\ \end{bmatrix} $$

robjohn
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  • Doesn't the initial state matrix have to be a column matrix, or am I wrong? – Mr Pie Aug 17 '21 at 03:56
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    You can transpose everything if you need it to be a column matrix, but there is no intrinsic reason it needs to be a column matrix. This can be formulated just as well with row vectors. – robjohn Aug 17 '21 at 04:09
  • Thanks for helping me clarify. – Mr Pie Aug 23 '21 at 23:32