When a matrix $A_0$, for example $2 \times 2$ matrix, is symmetric positive definite, the following steps can compute the eigenvalues of $A_0$,
$A_k=R^T_k R_k$ (Cholesky Decomposition)
$A_{k+1}=R_k R^T_k$
where $k$ goes from $0$ to $\infty$ i.e., $A_1$ is calculated using cholesky factor of $A_0$, $A_2$ is calculated using cholesky factor of $A_1$ and so on..
Finally, \begin{equation} A_\infty=\begin{bmatrix} \lambda_1&0\\ 0&\lambda_2 \end{bmatrix} \end{equation} where $\lambda_1$ and $\lambda_2$ are eigenvalues of $A_0$.Why this procedure can reach the eigenvalues of $A_0$?