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I've obtained a necessary and sufficient condition for the unshifted QR algorithm to converge. The condition for the algorithm to converge for a square matrix $M$ is:

For any two eigenvalues $\lambda$ and $\mu$ of $M$, if $|\lambda| = |\mu|$ then $\lambda = \mu$.

The condition implies that (a) The algorithm converges for all positive-definite Hermitian matrices (b) The algorithm converges for almost all square matrices over $\mathbb C$.

I'm guessing that this condition has already been proved. I'd like to compare my proof to other's. Where can I find a reference?

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See page 517 of Wilkinson's The Algebraic Eigenvalue Problem. The case when some eigenvalues may have equal modulus is treated on page 520.

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  • $\begingroup$ I'm confused. If we have that $|\lambda| = |\mu|$ while $\lambda \neq \mu$ then the algorithm does not convergence. For instance, take $M = \left[\begin{matrix}0 & 1\\1 & 0\end{matrix}\right]$ $\endgroup$
    – wlad
    Commented Oct 2, 2021 at 10:39
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    $\begingroup$ there remain nonzero subdiagonal elements when some eigenvalues have equal modulus; in particular, as discussed on page 521, when there are complex conjugate eigenvalues the convergence is to a matrix that has $2\times 2$ matrices centered on the diagonal with eigenvalues equal to that complex conjugate pair. $\endgroup$
    – Carlo Beenakker
    Commented Oct 2, 2021 at 12:24

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