What is the convergence and how to find eigenpairs in the QR Algorithm?

Question

I understand the basic QR algorithm which is, for given matrix A,

$A_{k} = A$ for k = 0,

$A_{k} = Q_{k}R_{k}$

$A_{k+1} = R_{k}Q_{k}$

repeat iteratively for k > 0

The algorithm supposedly finds the eigenvalues and eigenvectors of A but

1) what is the convergence and what's the condition and when to stop the iteration?

2) suppose after such convergence, $A_{k}$ is decomposed into $Q_{k}R_{k}$, then how can I figure out all of the eigenvalues and eigenvectors?

Answer 1

All of your questions are answered in the wiki on the QR Algorithm. An example of a specific convergence rate can be found here for symmetric matrices: QR-algorithm - convergence property

Your second question is superfluous. $A_k$ converges to a tridiagonal matrix where the diagonal entries are the eigenvalues of your original matrix $A$.