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Complex dot product
Complex coordinate space
nosplit
words:
87
math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169
The complex dot product is defined as:
∑
a
i
b
i
(11)
E.g. in
C
1
:
(
a
+
b
i
)
⋅
(
c
+
d
i
)
=
(
a
+
b
i
)
(
c
+
d
i
)
=
(
a
+
b
i
)
(
c
−
d
i
)
=
(
a
c
+
b
d
)
+
(
b
c
−
a
d
)
i
(12)
We can see therefore that this is a
form
, and a positive definite because:
(
a
+
b
i
)
⋅
(
a
+
b
i
)
=
(
a
a
+
b
b
)
+
(
b
a
−
a
b
)
i
=
a
2
+
b
2
(13)
Just like the usual
dot product
, this will be a
positive definite symmetric bilinear form
by definition.
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