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Unformatted text preview: t A∞ is normal. From last lecture’s proposition of QR algorithm we know
that An ’s are all unitarily equivalent. It can be checked that any matrix unitarily equivalent to a
normal matrix is also normal: say X and Y are unitarily equivalent, and X is normal. Then,
Y ∗ Y = (U ∗ XU )∗ (U ∗ XU )
= U ∗ X ∗ U U ∗ XU
= U ∗ X ∗ XU
= U ∗ XX ∗ U
= U ∗ XU U ∗ X ∗ U
= Y Y ∗,
and thus Y is normal. Hence, since An ’s are unitarily equivalent to A0 = A, and A is normal,
and thus the limiting matrix A∞ is also normal.)
By normality of R∞ and the fact that it is triangular, R∞ = D∞ with D∞ a diagonal matrix and
A ∞ = D∞ Q ∞ = D∞ Q ∞ .
Since {Q∞ , D∞ } is a commuting family in Mn , there exists a unitary U ∈ Mn that diagonalizes
both:
U ∗ D∞ U = D∞ , and U ∗ Q∞ U = Q∞ ,
where Q∞ is a diagonal matrix. We will denote the i’th diagonal entry of of Q∞ by ωi . Note
that ωi  = 1, since the eigenvalues of an unitary matrix are ±1, and the eigenvalues of Q∞ are
the diagonal entries of Q∞ . Hence, we obtain the following result:
U ∗ D∞ Q∞ U = U ∗ D∞ U U ∗ Q∞ U
= D∞ Q ∞ d11 0 . . . 0
ω1 0 . . . 0
. .
.
. .
. 0 d22 . . . 0 ω2 . . . = . . . . ..
..
..
..
.
. 0 .
. 0
.
.
0 . . . 0 dnn
0 . . . 0 ωn d11 ω1
0
...
0
.
. .
d22 ω2 . .
.
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
 Addition, Matrices

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