CS205 Homework #6
Problem 1
1. Let
A
be a symmetric and positive definite
n
×
n
matrix. If
x
,
y
∈
R
n
prove that the
operation
x
,
y
A
=
x
T
Ay
=
x
·
Ay
is an inner product on
R
n
. That is, show that the
following properties are satisfied
(a)
u
+
v
,
z
A
=
u
,
z
A
+
v
,
z
A
(b)
α
u
,
v
A
=
α
u
,
v
A
(c)
u
,
v
A
=
v
,
u
A
(d)
u
,
u
A
≥
0 and equality holds if and only if
u
=
0
2. Which of those properties, if any, fail to hold when
A
is not positive definite? Which
fail to hold if it is not symmetric?
Problem 2
1. Let
x
1
,
x
2
, . . . ,
x
k
be an
A
orthogonal set of vectors, that is
x
T
i
Ax
j
= 0 for
i
=
j
.
Show that if
A
is symmetric and positive definite, then
x
1
,
x
2
, . . . ,
x
k
are linearly
independent. Does this hold when
A
is symmetric but not positive definite?
2. Let
x
1
,
x
2
, . . . ,
x
n
be
n
linearly independent vectors of
R
n
and
A
a
n
×
n
symmetric
positive definite matrix. Show that we can use the GramSchmidt algorithm to create
a
full
A
orthogonal set of
n
vectors. That is, subtracting from
x
i
its
A
overlap with
x
1
,
x
2
, . . . ,
x
i

1
will never create a zero vector.
Problem 3
Let
A
be a
n
×
n
symmetric positive definite matrix. Consider the steepest descent method
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 Fall '07
 Fedkiw
 Linear Algebra, Gradient descent, positive definite

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