CS205 Homework #6 Solutions
Problem 1
1. Let
A
be a symmetric and positive deﬁnite
n
×
n
matrix. If
x
,
y
∈
R
n
prove that the
operation
h
x
,
y
i
A
=
x
T
Ay
=
x
·
Ay
is an inner product on
R
n
. That is, show that the
following properties are satisﬁed
(a)
h
u
+
v
,
z
i
A
=
h
u
,
z
i
A
+
h
v
,
z
i
A
(b)
h
α
u
,
v
i
A
=
α
h
u
,
v
i
A
(c)
h
u
,
v
i
A
=
h
v
,
u
i
A
(d)
h
u
,
u
i
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?
Solution
1. (a)
h
u
+
v
,
z
i
A
= (
u
+
v
)
T
Az
=
u
T
Az
+
v
T
Az
=
h
u
,
z
i
A
+
h
v
,
z
i
A
(b)
h
α
u
,
v
i
A
= (
α
u
)
T
Av
=
α
(
u
T
Av
) =
α
h
u
,
v
i
A
(c)
h
u
,
v
i
A
=
u
T
Av
=
u
T
A
T
v
=
Au
·
v
=
v
·
Au
=
v
T
Au
=
h
v
,
u
i
A
by symmetry
(d)
h
u
,
u
i
A
=
u
T
Au
≥
0 if
u
6
= 0 by positive deﬁniteness and equality holds trivially
when
u
= 0.
2. Property (3) holds if and only if
A
is symmetric. Property (4) holds if and only if
A
is positive deﬁnite (by deﬁnition)
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
6
=
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 Gram-Schmidt algorithm to create
a
full
A