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Might be a wild intuition , I'd say the eigenvalues are the entries of the first row , and that the eigenvector coresponding to the $nth$ eigenvalue ,$k$ is made by adjoining a column of zeroes to the eigenvector coresponding to the eigenvector coresponding to the same eigenvalue for the first $n*n$ minor of the matrix .

Example :

for eigenvaue $1$ we take the matrix $\begin{pmatrix} 1 \end{pmatrix}$ ,the eigenvetor corresponding to $1$ is $\begin{pmatrix} 1 \end{pmatrix}$ , so we obtain $\begin{pmatrix} 1 \cr 0 \cr 0 \cr 0 \cr 0 \cr \vdots \end{pmatrix}$ as the first eigevector .

for eigenvaue $1/2$ we take the matrix $\begin{pmatrix} 1& 1/2 \cr 0 & 1/2\end{pmatrix}$ ,the eigenvetor corresponding to $1/2$ is $\begin{pmatrix} 1 \cr -1\end{pmatrix}$ , so we obtain $\begin{pmatrix} 1 \cr -1 \cr 0 \cr 0 \cr 0 \cr \vdots \end{pmatrix}$ as the second eigevector .

The eigenvector coresponding to $1/8$ for $\begin{pmatrix} 1& 1/2 & 1/8 \cr 0 & 1/2 & 1/4 \cr 0 & 0 & 1/8 \end{pmatrix}$ is $\begin{pmatrix} 5 \cr -14 \cr 21 \cr \end{pmatrix}$, you get the ideea .Also , the eigenvectors span the entire space , ie if a possibly infinite (but convergent) sum of eigenvectors is $\vec 0$ then the coefficients of those vectors are $0$ .

Here is an explicit formula for the eigenvectors :first select $M_n$ , the $n*n$ truncation of the matrix and calculate $M_n - I*v_n$ , the nt'h eigenvalue . Example : for n=3 , we obtain \begin{pmatrix} 7/8 & 1/2 & 1/8 \cr 0 & 3/8 & 1/4 \cr 0 & 0 & 0 \end{pmatrix} . Now let $S$ be the $(n-1)*(n-1)$ truncation of that , ie \begin{pmatrix} 7/8 & 1/2 \cr 0 & 3/8 & \end{pmatrix} Calculate $S^{-1}$ = \begin{pmatrix} 8/7 & -32/21 \cr 0 & 8/3 & \end{pmatrix} , now multiply $S^{-1}$ with the truncation of the last column of $M_n$ , \begin{pmatrix} 1/8 \cr 1/4 \end{pmatrix} You obtain \begin{pmatrix} -5/21 \cr 2/3 \cr \end{pmatrix} . Concatenating $-1$ to that , you obtain \begin{pmatrix} -5/21 \cr 2/3 \cr -1 \end{pmatrix} , the third eigenvector ,or the nt'h eigenvector in the general case .

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Might be a wild intuition , I'd say the eigenvalues are the entries of the first row , and that the eigenvector coresponding to the $nth$ eigenvalue ,$k$ is made by adjoining a column of zeroes to the eigenvector coresponding to the eigenvector coresponding to the same eigenvalue for the first $n*n$ minor of the matrix .

Example :

for eigenvaue $1$ we take the matrix $\begin{pmatrix} 1 \end{pmatrix}$ ,the eigenvetor corresponding to $1$ is $\begin{pmatrix} 1 \end{pmatrix}$ , so we obtain $\begin{pmatrix} 1 \cr 0 \cr 0 \cr 0 \cr 0 \cr \vdots \end{pmatrix}$ as the first eigevector .

for eigenvaue $1/2$ we take the matrix $\begin{pmatrix} 1& 1/2 \cr 0 & 1/2\end{pmatrix}$ ,the eigenvetor corresponding to $1/2$ is $\begin{pmatrix} 1 \cr -1\end{pmatrix}$ , so we obtain $\begin{pmatrix} 1 \cr -1 \cr 0 \cr 0 \cr 0 \cr \vdots \end{pmatrix}$ as the second eigevector .

The eigenvector coresponding to $1/8$ for $\begin{pmatrix} 1& 1/2 & 1/8 \cr 0 & 1/2 & 1/4 \cr 0 & 0 & 1/8 \end{pmatrix}$ is $\begin{pmatrix} 5 \cr -14 \cr 21 \cr \end{pmatrix}$, you get the ideea .Also , the eigenvectors span the entire space , ie if a possibly infinite (but convergent) sum of eigenvectors is $\vec 0$ then the coefficients of those vectors are $0$ .