ECE 493
HW #2 –
Version 1.00 February 24, 2011
Spring 2011
Univ. of Illinois
Due Thur, Mar. 10
Prof. Allen
Topic of this homework:
LinearAlgebra(Cauchy Schwartz inequality for n dimensional space.
Eigenvalues and eigenvectors)
Deliverables: Show your work. Numerical results are not suFcient expect when speci±cally
requested.
1. Prove the Cauchy Schwartz inequality by following the argument below.
(a) Use the argument based on the proof on pages 424426 in the text book. However use
the following. De±ne
E
(
a
) =

V

aU

and then ±nd the minimum of
E
with respect
to
a
. As shown in the book, use this to prove the CS inequality (13) page 424.
(b) Give a diagram showing what is happening when you minimize
E
(
a
) with respect to
a
.
Provide a diagram showing the minimum value of
E
(
a
*
) where
a
*
is the value such that
the gradient of
E
with respect to
a
is zero.
2. (a) Prove the triangular inequality

vu
+
v±
 ≤ 
vu

+

v±

. All of these are to be done for
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 Spring '11
 JontB.Allen
 Linear Algebra, Univ. of Illinois, Cauchy Schwartz, Cauchy Schwartz inequality

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