MATH 294
Chapter 5.5 Inner Product Spaces
Ex 5.5.3
Let S be a
n
×
n
matrix. In
R
n
deﬁne the product
< ~x,~
y >
= (
S~x
)
T
S~
y
.
Part a:
For which choices of S
is this an inner product?
Solution:
We know that an inner product must fulﬁll the following properties (deﬁnition
5.5.1):
(i)
< ~x,~
y >
=
< ~
y,~x >
(ii)
< ~x
1
+
~x
2
,~
y >
=
< ~x
1
,~
y >
+
< ~x
2
,~
y >
(iii)
< c~x,~
y >
=
c < ~x,~
y >,
∀
c
∈
R
(iv)
< ~x,~x > >
0
,
∀
~x
∈
R
n
such that
~x
6
=
~
0
Let’s present
< ~x,~
y >
in the following form
< ~x,~
y >
= (
S~x
)
T
S~
y
=
~x
T
S
T
S~
y
. If we denote
S
T
S
by B (notice that B
is symmetric) then
< ~x,~
y >
=
~x
T
B~
y
∀
~x,~
y
∈
R
n
. Let’s check (i)-(iv).
(i)
< ~x,~
y >
=
~x
T
B~
y
= (
~x
T
B~
y
)
T
=
~
y
T
B~x
=
< ~
y,~x >
√
(ii)
< ~x
1
+
~x
2
,~
y >
= (
~x
1
+
~x
2
)
T
B~
y
=
~x
T
1
B~
y
+
~x
T
2
B~
y
=
< ~x
1
,~
y >
+
< ~x
2
,~
y >
√
(iii)
< c~x,~
y >
=
c~x
T
B~
y
=
c < ~x,~
y >,
∀
c
∈
R
√
(iv)
< ~x,~x >
= (
S~x
)
T
S~x
= (let
~
z
=
S~x
) =
~
z
T
~
z
=
k
~
z
k
2
>
0
∀
~
z
∈
R
n
such that
~
z
6
=
~
0
So
~
z
=
S~x
must be diﬀerent from
~
0 for any
~x
6
=
~
0. It holds iﬀ the system of linear equations
S~x
=
~
0 has an unique
solution
~x
=
~
0. So we’re going to get (iv) holds
⇔
S is invertible. So
< ~x,~
y >
is inner product
⇔
S is invertible.
Part b: