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.