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Math 235
Assignment 7
Due: Wednesday, June 30th
1.
For each quadratic form
Q
(
~x
), determine the corresponding symmetric matrix
A
. By
diagonalizing
A
, Write
Q
so that it has no cross terms and give the change of variables
which brings it into this form. Classify each quadratic form as positive deﬁnite, negative
deﬁnite or indeﬁnite.
a)
Q
(
x,y
) =
x
2
+ 6
xy

7
y
2
.
b)
Q
(
x,y,z
) = 4
x
2
+ 4
xy
+ 4
y
2
+ 4
xz
+ 4
yz
+ 4
z
2
2.
Let
A
=
±

5 3

3 1
²
. Find an orthogonal matrix
P
and upper triangular matrix
T
such that
P
T
AP
=
T
.
3.
Let
Q
(
~x
) =
~x
T
A~x
with
A
=
±
a b
b c
²
and det
A
6
= 0.
a) Prove that
Q
is positive deﬁnite if det
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Unformatted text preview: A > 0 and a > 0. b) Prove that Q is negative deﬁnite if det A > 0 and a < 0. c) Prove that Q is indeﬁnite if det A < 0. 4. Let A be an invertible symmetric matrix. Prove that if the quadratic form ~x T A~x is positive deﬁnite, then so it the quadratic form ~x T A1 ~x . 5. Let A be an n × n symmetric matrix and let ~x,~ y ∈ R n . Deﬁne < ~x,~ y > = ~x T A~ y . Prove that < ,> is an inner product on R n if and only if A is positive deﬁnite....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Math

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