I understand that the key to retrieve eigenvectors in the non-shifted QR algorithm is to accumulate the transformations at each steps in the following way:


$Q = \Pi\_i Q\_i$


Can we accumulate the transformations in the same way when we are using the shifted QR algorithm?


$(A\_i-\mu I) = Q\_i R\_i$


$R\_i Q\_i + \mu I = A\_{i+1}$ 


If not, how can we retrieve the eigenvectors?



How to retrieve eigenvectors from shifted QR algorithm?
At each step, the shifted and unshifted matrices, $A-\lambda I$ and $A$, have the same eigenvectors. So the diagonalizing transformation $A = Q D Q^T$, with $D$ diagonal, is accumulated in the same way as in the unshifted algorithm, $Q = Q\_1 Q\_2 \cdots Q\_k$, where $k$ is the last step of the iteration.


