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Unformatted text preview: x = 0 ∀x ∈ Cn
Using this, we proceed with the proof:
(⇒) : If A is normal, then:
A∗ A = AA∗ .
This implies:
(A∗ A − AA∗ )x, x = 0, ∀x ∈ Cn
By the bilinearity of the inner product, this implies that:
A∗ Ax, x = AA∗ x, x, ∀x ∈ Cn
By the deﬁnition of the adjoint and the fact that A = (A∗ )∗ , this implies:
Ax2 = Ax, Ax = A∗ Ax, x = AA∗ x, x = A∗ x, A∗ x = A∗ x2 , ∀x ∈ Cn
Hence:
Ax = A∗ x, ∀x ∈ Cn
(⇐) : Suppose Ax = A∗ x ∀x ∈ Cn .
Then by following the logic from the (⇒) direction backwards, we can get back to:
(A∗ A − AA∗ )x, x = 0, ∀x ∈ Cn .
Now we may use the Polarization Identity along with some rearranging to conclude that :
(A∗ A − AA∗ )x, y = 0, ∀x, y ∈ Cn .
Hence:
A∗ A − AA∗ = 0 (refer to today’s clariﬁcation)
Thus, A is normal.
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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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