QR Factorization for 2x2 does not give correct Eigenvalue To find the Eigenvalue as per QR Factorization, the diagonal values of R are the EigenValues while every column in Q is an EigenVector. However, this does not seem to be the case in a 2x2 matrix.
Consider M = \begin{bmatrix}2&1\\1&2\end{bmatrix}
Then the QR decomposition (see http://comnuan.com/cmnn0100e/) yields
Q = \begin{bmatrix}.89&-.44\\-.44&-.89\end{bmatrix}
R = \begin{bmatrix}-2.23&-1.78\\0&1.34\end{bmatrix} 
Note, QR = M checks out.
Note as per this the EigenValues are -2.23 and 1.34, while if you do the calculation manually it is easy to see that EigenValues for M are 1 and 3.
More importantly, M.C = $\lambda$.C does not come correct using the QR method. In our case. for $\lambda$ = -2.23, 
C =  \begin{bmatrix}.89\\-.44\end{bmatrix}
but this does not satisfy: 
M.C = $\lambda$.C
Is QR not suitable for 2x2 matrices, or is my understanding wrong ?
 A: Based on what you’ve said in the comments to your question, it looks like you might be confusing the QR algorithm for finding eigenvalues of a matrix with the QR decomposition of a matrix.  
Computing the QR decomposition of a matrix is a step in the inner loop of the QR algorithm. In the simplest version of this algorithm, after computing the matrices $Q$ and $R$, you then replace your matrix with $RQ$ and loop. If this process converges, then $R$ converges to a diagonal matrix with the eigenvalues of the original matrix along the diagonal.
So, in computing the QR decomposition of $M$, you’ve only performed part of one iteration of this algorithm. The matrix $R$ that resulted from this isn’t even close to being diagonal, so there’s no reason to expect that its main diagonal elements are the eigenvalues of $M$. If you now compute $RQ$, compute the QR decomposition of that, &c., after a half-dozen iterations or so you’ll see $R$ begin to converge to a diagonal matrix with the values that you expect.
