Definition: Inner Product
You may have run across
inner products, also
called
dot products, on
ℝn
n
before in some of your math or science courses. If
not, we define the inner product as follows, given we have some
x∈ℝn
x
n
and
y∈ℝn
y
n
Definition 1:
inner product
The inner product is defined mathematically as:
<x,y>=yTx=
y
0
y
1
…
y
n
−
1
x
0
x
1
⋮
x
n
−
1
=∑i=0n-1
x
i
y
i
x
y
y
x
y
0
y
1
…
y
n
−
1
x
0
x
1
⋮
x
n
−
1
i
n
1
0
x
i
y
i
(1)
Inner Product in 2-D
If we have
x∈ℝ2
x
2
and
y∈ℝ2
y
2
, then we can write the inner product as
<x,y>=∥x∥∥y∥cosθ
x
y
x
y
θ
(2)
where
θθ is the angle
between
xx and
yy.
Geometrically, the inner product tells us about the
strength of xx in the direction of
yy.
Example 1
For example, if
∥x∥=1
x
1
, then
<x,y>=∥y∥cosθ
x
y
y
θ
The following characteristics are revealed by the inner
product:
-
<x,y>
x
y
measures the length of the
projection of yy onto
xx.
-
<x,y>
x
y
is maximum (for given
∥x∥
x
,
∥y∥
y
)
when xx
and yy are
in the same direction (
θ=0⇒cosθ=1
θ
0
θ
1
).
-
<x,y>
x
y
is zero when
cosθ=0⇒θ=90°
θ
0
θ
90°
, i.e. xx and yy are
orthogonal.
Inner Product Rules
In general, an inner product on a complex vector space is
just a function (taking two vectors and returning a complex
number) that satisfies certain rules:
-
Conjugate Symmetry:
<x,y>=<x,y>¯
x
y
x
y
-
Scaling:
<αx,y>=α<x,y>
α
x
y
α
x
y
-
Additivity:
<x+y,z>=<x,z>+<y,z>
x
y
z
x
z
y
z
-
"Positivity":
∀x,x≠0:<x,x>>0
x
x
0
x
x
0
Definition 2:
orthogonal
We say that xx
and yy are
orthogonal if:
<x,y>=0
x
y
0
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