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Unformatted text preview: o A
Proof. By deﬁnition, An+1 = Rn Qn = Q∗ An Qn since An = Qn Rn implies that Rn = Q∗ An .
So An+1 and An are unitarily equivalent. Thus, by induction An and A0 are as well. 3 Thus, if An were to converge (in the sense of entry-wise convergence) to an upper triangular
matrix, then by unitary equivalence we would know all eigenvalues of A. Problem: QR algorithm
may not converge. For example,
= Q 0 R0
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