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My question is related to the derivation of eigenvectors of a transition probability matrix in Hamilton's ''Time Series Analysis''. I have troubles deriving the same eigenvectors as what the author provides in the book. The problem is as follows:

Let

${\mathbf P}=\left[ \begin{array}{cc} p_{11} & 1-p_{22} \\ 1-p_{11} & p_{22} \end{array} \right]. $

Its eigenvalues can be found from $\left|P-\lambda I_N\right|=0$ and they are ${\lambda }_1=1$ and ${\lambda }_2=-1+p_{11}+p_{22}$. Next the author concludes that the eigenvector related to ${\lambda }_1=1$ is $\ \left[ \begin{array}{c} {(1-p_{22})}/{(2-p_{11}-p_{22})} \\ {(1-p_{11})}/{(2-p_{11}-p_{22})} \end{array} \right]$

and the one related to ${\lambda }_2=-1+p_{11}+p_{22}$ is $\left[ \begin{array}{c} -1 \\ 1 \end{array} \right].$

We get${\mathbf \ }{\mathbf T}{\mathbf =}\left[ \begin{array}{cc} \frac{1-p_{22}}{2-p_{11}-p_{22}} & -1 \\ \frac{1-p_{11}}{2-p_{11}-p_{22}} & 1 \end{array} \right]$ and ${\mathbf \Lambda }{\mathbf =}\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1+p_{11}+p_{22} \end{array} \right].$

It's true that it holds ${\mathbf P}{\mathbf =}{\mathbf T}{\mathbf \Lambda }{{\mathbf T}}^{{\rm -}{\rm 1}}$.

On the other hand when I derive the eigenvectors through $(P-\lambda I)t=0$, I get different eigenvectors, particularly $\left[ \begin{array}{cc} 1-p_{22} & -1 \\ 1-p_{11} & 1 \end{array} \right]$. And as a result $\left[ \begin{array}{cc} 1-p_{22} & -1 \\ 1-p_{11} & 1 \end{array} \right]\ne \ \left[ \begin{array}{cc} \frac{1-p_{22}}{2-p_{11}-p_{22}} & -1 \\ \frac{1-p_{11}}{2-p_{11}-p_{22}} & 1 \end{array} \right] $

So my question is why $2-p_{11}-p_{22}$ is in the denominator of an eigenvector derived by the author?

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First, there is no such thing as “the” eigenvector, since any non-zero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.

Clearly, the eigenvector of $1$ that you found and the one the author uses are scalar multiples of each other, so they are equivalent. The author has normalized it so that the sum of its components is $1$, i.e., so that it’s a probability vector that turns out to be the limiting state of the process represented by the matrix.

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