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I'm currently getting most of my info from Burden's Numerical Analysis book . In the book it mentions that the $QR$ algorithm converges to a diagonal matrix with no proof provided. The method in the book consists of converting a symmetric matrix $A$ to a congruent tridiagonal matrix using householder reflectors then using rotational matrices to find the $QR$ factorization and then proceeding with the algorithm.I investigated further online and am aware that unshifted $QR$ only converges assuming all eigenvalues are distinct. The only proof i have found so far is from this page http://pi.math.cornell.edu/~web6140/TopTenAlgorithms/QRalgorithm.html. But i haven't seen eigenvslue decomposition yet and certain parts of that proof are confusing. I wanted to ask if there was a simpler proof someone here might know or if you can direct me to a proof.

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  • $\begingroup$ I know that. I'm aware of what $Q$ and $R$ are. Everything i have read so far says that it does converge to a diagonal matrix , but if the unshifted $QR$ algorithm doesn't converge to a diagonal matrix, what exactly does it converge to then? $\endgroup$ – Javi maxwell May 5 at 4:48
  • $\begingroup$ Are you taking a class in numerical linear algebra. See trefethan and bau it's lecture 28.,pg 217. See google books. Thought you were talking about QR factorization at first. $\endgroup$ – Shogun May 5 at 4:54
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I'm going to close this question, I was able to find a proof. If anybody is interested, i found the proof on "The Algebraic Eigenvalue Problem" by J.H. Wilkinson

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