# QR alogrithm for eigenvalue problem [closed]

Considering pure QR algorithm (without shifts and preliminary tridiagonal reduction) are there sufficient conditions for algorithm to converge to quasi-diagonal form?

For the the following matrix $$A = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right).$$ with eigenvalues $\lambda_1 = 1$ and $\lambda_2 = -1$ it apparently does not. QR decomposition produces the following $$A = QR = \left(\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right) \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right),$$ which generates stationary sequence $\{A^{(k)}\}$.

• You didn't miss anything. Maybe you just missunderstood what is known about the QR-algorithm. There is no general convergence statement for the QR-algorithm, only theorems like "for almost all inputs it converges". You just found an example that shows you that "almost all" is as good as is gets... Sep 9 '13 at 16:25
• Shifts. You're missing shifts. Check out the document in @guest's answer. Sep 9 '13 at 16:29