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Unformatted text preview: ·∗
λ1 ∗ · · · ∗
0
0 ∗
= U2 . U2 = .
.
.
.
.
0
0 λ1 ∗ · · · ∗
. = 0 = T,
.
.
0 )∗BU (U B T which is upper triangular with λ1 , λ2 , ..., λn along its diagonal. This completes the proof, since
A = (U ∗ ) ∗ T U ∗ . 2.2 Unitary diagonalization for normal matrices Now we will examine the consequences of this theorem. We’ll begin by ﬁnding a condition for
unitary diagonalizability.
2.2.2 Deﬁnition. A matrix A ∈ Mn is called normal if:
AA∗ = A∗ A
.
2.2.3 Remark. Hermitian and unitary matrices are normal, but there are normal matrices which
are neither Hermitian nor unitary.
1 −1
2.2.4 Example. A =
is normal, but it is neither Hermitian nor unitary.
11
2.2.5 Proposition. A matrix A ∈ Mn is normal iﬀ Ax = A∗ x, for all x ∈ Cn .
3 Proof. First note that:
A∗ A = AA∗ ⇔ A∗ A − AA∗ = 0 ⇔ (A∗ A − AA∗ )...
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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
 Matrices

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