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In der linearen Algebra ist die Vektor- und Matrixdifferenzierung oder auch kurz Matrixdifferenzierung die Ableitung von Matrizen und Vektoren.
Sei die Form
y
=
A
x
{\displaystyle \mathbf {y} =\mathbf {A} \mathbf {x} }
A
{\displaystyle \mathbf {A} }
m
×
n
{\displaystyle m\times n}
x
{\displaystyle x}
n
{\displaystyle n}
∂
y
∂
x
=
A
{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}=\mathbf {A} }
Sei
y
=
z
⊤
A
x
{\displaystyle \mathbf {y} =\mathbf {z} ^{\top }\mathbf {A} \mathbf {x} }
z
{\displaystyle \mathbf {z} }
m
×
1
{\displaystyle m\times 1}
A
{\displaystyle \mathbf {A} }
m
×
n
{\displaystyle m\times n}
x
{\displaystyle \mathbf {x} }
n
×
1
{\displaystyle n\times 1}
z
{\displaystyle \mathbf {z} }
x
{\displaystyle \mathbf {x} }
∂
y
∂
z
=
x
⊤
A
⊤
{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {z} }}=\mathbf {x} ^{\top }\mathbf {A} ^{\top }}
und
∂
y
∂
x
=
z
⊤
A
{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}=\mathbf {z} ^{\top }\mathbf {A} }
[ 1] Es sei die quadratische Form
y
=
z
⊤
A
z
{\displaystyle \mathbf {y} =\mathbf {z} ^{\top }\mathbf {A} \mathbf {z} }
A
{\displaystyle \mathbf {A} }
n
×
m
{\displaystyle n\times m}
z
{\displaystyle \mathbf {z} }
n
{\displaystyle n}
z
{\displaystyle \mathbf {z} }
∂
z
⊤
A
z
∂
z
=
z
⊤
(
A
+
A
⊤
)
{\displaystyle {\frac {\partial \mathbf {z} ^{\top }\mathbf {A} \mathbf {z} }{\partial \mathbf {z} }}=\mathbf {z} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}
Beweis :
Per Definition gilt:
y
=
∑
j
=
1
n
∑
i
=
1
n
a
i
j
z
i
z
j
{\displaystyle \mathbf {y} =\sum _{j=1}^{n}\sum _{i=1}^{n}a_{ij}z_{i}z_{j}}
Wenn man nach dem
k
{\displaystyle k}
∂
y
∂
z
k
=
∑
j
=
1
n
a
k
j
z
j
+
∑
i
=
1
n
a
i
k
z
i
k
=
1
,
…
,
n
{\displaystyle {\frac {\partial \mathbf {y} }{\partial z_{k}}}=\sum _{j=1}^{n}a_{kj}z_{j}+\sum _{i=1}^{n}a_{ik}z_{i}\quad k=1,\ldots ,n}
und damit schließlich
∂
y
∂
z
=
z
⊤
A
⊤
+
z
⊤
A
=
z
⊤
(
A
+
A
⊤
)
{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {z} }}=\mathbf {z} ^{\top }\mathbf {A} ^{\top }+\mathbf {z} ^{\top }\mathbf {A} =\mathbf {z} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}
Falls
A
{\displaystyle \mathbf {A} }
∂
z
⊤
A
z
∂
z
=
2
z
⊤
A
{\displaystyle {\frac {\partial \mathbf {z} ^{\top }\mathbf {A} \mathbf {z} }{\partial \mathbf {z} }}=2\mathbf {z} ^{\top }\mathbf {A} }
Für Ableitungen von Determinanten gilt es vielfältige Rechenregeln. Die einfachste ist die Ableitung einer Matrix nach sich selbst. Sei A eine quadratische Matrix der Ordnung m, dann gilt für ihre Ableitung ihrer Determinante
∂
|
A
|
∂
A
=
A
∗
{\displaystyle {\frac {\partial |\mathbf {A} |}{\partial \mathbf {A} }}=\mathbf {A} ^{*}}
wobei
A
∗
{\displaystyle \mathbf {A} ^{*}}
Kofaktormatrix ist.
Sei
A
{\displaystyle \mathbf {A} }
m
×
n
{\displaystyle m\times n}
X
{\displaystyle \mathbf {X} }
n
×
m
{\displaystyle n\times m}
B
{\displaystyle \mathbf {B} }
m
×
m
{\displaystyle m\times m}
∂
Spur
(
A
X
B
)
∂
X
=
A
⊤
B
⊤
{\displaystyle {\frac {\partial \operatorname {Spur} (\mathbf {A} \mathbf {X} \mathbf {B} )}{\partial \mathbf {X} }}=\mathbf {A} ^{\top }\mathbf {B} ^{\top }}
↑ Phoebus J. Dhrymes: Mathematics for Econometrics . 4 Auflage. S. 159 ff.
Kategorie:Lineare Algebra
