Het standaardinproduct
⟨
x
,
y
⟩
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle }
vectoren
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1
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x
2
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x
n
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,
y
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y
1
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y
2
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∈
R
n
{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n}),\ \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n}}
⟨
x
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y
⟩
=
x
1
y
1
+
x
2
y
2
+
…
+
x
n
y
n
=
∑
i
=
1
n
x
i
y
i
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =x_{1}y_{1}+x_{2}y_{2}+\ldots +x_{n}y_{n}=\sum _{i=1}^{n}x_{i}y_{i}}
Vat men
x
{\displaystyle \mathbf {x} }
y
{\displaystyle \mathbf {y} }
kolomvectoren :
x
=
[
x
1
,
x
2
,
…
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x
n
]
⊤
,
y
=
[
y
1
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y
2
,
…
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y
n
]
⊤
{\displaystyle \mathbf {x} =[x_{1},x_{2},\ldots ,x_{n}]^{\top },\mathbf {y} =[y_{1},y_{2},\ldots ,y_{n}]^{\top }}
dan kan het standaardinproduct als matrixproduct worden geschreven:
⟨
x
,
y
⟩
=
x
⊤
y
=
y
⊤
x
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} ^{\top }\mathbf {y} =\mathbf {y} ^{\top }\mathbf {x} }
Van het complexe standaardinproduct van twee vectoren
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x
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,
y
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)
∈
C
n
{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n}),\mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {C} ^{n}}
⟨
x
,
y
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=
x
¯
1
y
1
+
x
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2
y
2
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⋯
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x
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n
y
n
=
∑
i
=
1
n
x
¯
i
y
i
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle ={\bar {x}}_{1}y_{1}+{\bar {x}}_{2}y_{2}+\dotsb +{\bar {x}}_{n}y_{n}=\sum _{i=1}^{n}{\bar {x}}_{i}y_{i}}
en
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y
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′
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⋯
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x
n
y
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n
=
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i
=
1
n
x
i
y
¯
i
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle '=x_{1}{\bar {y}}_{1}+x_{2}{\bar {y}}_{2}+\dotsb +x_{n}{\bar {y}}_{n}=\sum _{i=1}^{n}x_{i}{\bar {y}}_{i}}
De beide vormen verschillen alleen daarin dat ze elkaars complex geconjugeerde zijn:
⟨
x
,
y
⟩
′
=
⟨
x
,
y
⟩
¯
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle '={\overline {\langle \mathbf {x} ,\mathbf {y} \rangle }}}
Vat men
x
{\displaystyle \mathbf {x} }
y
{\displaystyle \mathbf {y} }
x
=
[
x
1
,
x
2
,
…
,
x
n
]
⊤
,
y
=
[
y
1
,
y
2
,
…
,
y
n
]
⊤
{\displaystyle \mathbf {x} =[x_{1},x_{2},\ldots ,x_{n}]^{\top },\mathbf {y} =[y_{1},y_{2},\ldots ,y_{n}]^{\top }}
dan kan het complexe standaardinproduct geschreven worden als matrixproduct:
⟨
x
,
y
⟩
=
x
⊤
y
¯
=
y
¯
⊤
x
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\mathbf {x} ^{\top }{\bar {\mathbf {y} }}={\bar {\mathbf {y} }}^{\top }\mathbf {x} }
Het reële standaardinproduct heeft de eigenschap dat voor iedere reële vierkante matrix
A
{\displaystyle \mathbf {A} }
dimensie van de vectorruimte , geldt:
⟨
A
x
,
y
⟩
=
(
A
x
)
⊤
y
=
x
⊤
A
⊤
y
=
⟨
x
,
A
⊤
y
⟩
{\displaystyle \langle \mathbf {Ax} ,\mathbf {y} \rangle =(\mathbf {Ax} )^{\top }\mathbf {y} =\mathbf {x} ^{\top }\mathbf {A} ^{\top }\mathbf {y} =\langle \mathbf {x} ,\mathbf {A} ^{\top }\mathbf {y} \rangle }
Voor het complexe standaardinproduct geldt een soortgelijke de eigenschap, namelijk dat voor iedere complexe vierkante matrix
A
{\displaystyle \mathbf {A} }
⟨
A
x
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y
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=
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A
x
)
⊤
y
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=
x
⊤
A
⊤
y
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x
⊤
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⊤
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x
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⊤
y
⟩
{\displaystyle \langle \mathbf {\mathbf {Ax} } ,\mathbf {y} \rangle =(\mathbf {Ax} )^{\top }{\bar {\mathbf {y} }}=\mathbf {x} ^{\top }\mathbf {A} ^{\top }{\bar {\mathbf {y} }}=\mathbf {x} ^{\top }{\overline {{\bar {\mathbf {A} }}^{\top }\mathbf {y} }}=\langle \mathbf {x} ,{\bar {\mathbf {A} }}^{\top }\mathbf {y} \rangle }
De norm van een reële of complexe vector
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{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})}
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|
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|
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2
{\displaystyle \|\mathbf {x} \|={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle \ }}={\sqrt {|x_{1}|^{2}+|x_{2}|^{2}+\ldots +|x_{n}|^{2}\ }}}
De euclidische afstand
‖
x
−
y
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{\displaystyle \|\mathbf {x} -\mathbf {y} \|}
x
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{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})}
y
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)
{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})}
stelling van Pythagoras afgeleid:
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=
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x
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{\displaystyle \|\mathbf {x} -\mathbf {y} \|={\sqrt {\langle \mathbf {x} -\mathbf {y} ,\mathbf {x} -\mathbf {y} \rangle \ \ }}={\sqrt {|x_{1}-y_{1}|^{2}+\ldots +|x_{n}-y_{n}|^{2}\ }}}
De hoek
θ
=
∠
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x
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y
)
{\displaystyle \theta =\angle (\mathbf {x} ,\mathbf {y} )}
x
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≠
0
{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\neq 0}
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≠
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{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\neq 0}
cosinus van het reële standaardinproduct afgeleid:
cos
(
θ
)
=
⟨
x
,
y
⟩
‖
x
‖
‖
y
‖
{\displaystyle \cos(\theta )={\frac {\langle x,y\rangle }{\|x\|\ \|y\|}}}
Twee reële of complexe vectoren
x
=
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x
1
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x
2
,
…
,
x
n
)
≠
0
{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})\neq 0}
y
=
(
y
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y
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,
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y
n
)
≠
0
{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\neq 0}
⟨
x
,
y
⟩
=
0
{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0}
In het geval van dat twee reële vectoren
x
{\displaystyle \mathbf {x} }
y
{\displaystyle \mathbf {y} }
∠
(
x
,
y
)
=
90
∘
{\displaystyle \angle (\mathbf {x} ,\mathbf {y} )=90^{\circ }}
Het inwendige product van twee vectoren
x
=
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x
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x
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n
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{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})}
y
=
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y
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y
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,
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n
)
{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})}
n
{\displaystyle n}
V
{\displaystyle \mathbf {V} }
lineaire combinatie van een orthonormale basis
{
e
1
,
…
,
e
n
}
{\displaystyle \{\mathbf {e} _{1},\ldots ,\mathbf {e} _{n}\}}
V
{\displaystyle \mathbf {V} }
x
=
∑
i
=
1
n
x
i
e
i
{\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {e} _{i}}
y
=
∑
i
=
1
n
y
i
e
i
{\displaystyle \mathbf {y} =\sum _{i=1}^{n}y_{i}\mathbf {e} _{i}}
is gelijk aan:
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⟨
∑
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{\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =\left\langle \sum _{i=1}^{n}x_{i}\mathbf {e} _{i},\sum _{j=1}^{n}y_{j}\mathbf {e} _{j}\right\rangle =\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}y_{j}\langle \mathbf {e} _{i},\mathbf {e} _{j}\rangle =\sum _{i=1}^{n}x_{i}y_{i}}
![{\displaystyle \mathbf {x} =[x_{1},x_{2},\ldots ,x_{n}]^{\top },\mathbf {y} =[y_{1},y_{2},\ldots ,y_{n}]^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7fbc6f41607f97cd3e49913285f2ee9eacf5d73)
