# Computing eigenvectors from the QR algorithm

I've seen a few other posts on this topic but none have full answers.

I'm trying to implement some eigen-decomposition algorithms. I've managed to get the Explicit QR algorithm and the Implicit (Francis) algorithm to compute the eigenvalues. My question however relates to computing the eigenvectors.

My core question is:

Do I need to use the Inverse Iteration Algorithm to get the eigenvectors after completing the QR algorithm?

I understand additionally that I'll need to use the transformation matrices from my QR algorithm (balancing, the hessenberg reduction and the QR decomposition). But I'm more interested right now in whether there is some way to get the eigenvectors from the QR algorithm itself (as wikipedia implies there is).

• For reference I've been using the following sources: 1. people.inf.ethz.ch/arbenz/ewp/Lnotes/chapter3.pdf - Implementation of Hessenberg and implicit QR algorithm. 2. On Matrix Balancing and EigenVector Computation - James, Langou and Lowery. – user124784 Jan 31 '16 at 21:50
• If the matrix is normal, then the columns of the $Q$ matrix are precisely the eigenvectors. – Christian Clason Jan 31 '16 at 22:28
• @ChristianClason This covers symmetric matrices. Is there a way to recover the eigenvectors for general matrices? – user124784 Jan 31 '16 at 22:50
• Well, normal matrices, which is more general than symmetric. General (non-normal) matrices need not have eigenvectors (in the sense of unique vectors up to scalar multiples). – Christian Clason Feb 1 '16 at 0:10
• If your matrix has distinct real eigenvectors, they will be the columns of the $Q$ matrix. (And the $R$ matrix will be diagonal.) This is the case if and only if the matrix is normal. – Christian Clason Feb 1 '16 at 0:18