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Proof. By deﬁnition, An+1 = Rn Qn = Q∗ An Qn since An = Qn Rn implies that Rn = Q∗ An .
n
n
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 entrywise 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,
01
01
20
A0 =
=
= Q 0 R0
20
10
01
20
0...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.
 Fall '12
 BernhardBodmann
 Math, Matrices

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