Die geränderte Hesse-Matrix (engl. bordered Hessian ) dient zur Klassifikation von stationären Punkten bei mehrdimensionalen Extremwertproblemen mit Nebenbedingungen. Sie ist mit der „normalen“ Hesse-Matrix verwandt. Im Gegensatz zur Hesse-Matrix, welche auf positive bzw. negative Definitheit untersucht wird, ist bei der geränderten Hesse-Matrix das Vorzeichen der Determinante entscheidend.
Entscheidend ist die Vorzeichenfolge der führenden Hauptminoren , wobei gilt, dass man lediglich die k führenden Hauptminoren untersucht, für die gilt:
k
>
2
m
{\displaystyle k>2m}
k
>
2
⋅
1
→
k
>
2
{\displaystyle k>2\cdot 1\rightarrow k>2}
Sei
U
⊂
R
n
{\displaystyle U\subset \mathbb {R} ^{n}}
f
:
U
→
R
{\displaystyle f:U\rightarrow \mathbb {R} }
a
∈
U
{\displaystyle a\in U}
F
=
0
{\displaystyle F=0}
F
=
(
F
1
,
…
,
F
m
)
:
U
→
R
m
{\displaystyle F=(F_{1},\ldots ,\,F_{m}):U\rightarrow \mathbb {R} ^{m}}
m
<
n
{\displaystyle m<n}
L
(
λ
1
,
…
,
λ
m
,
x
)
:=
f
(
x
)
−
∑
i
=
1
m
λ
i
F
i
(
x
)
{\displaystyle L(\lambda _{1},\ldots ,\,\lambda _{m},\,x):=f(x)-\sum _{i=1}^{m}\lambda _{i}F_{i}(x)}
die Lagrange-Funktion mit der Abkürzung
x
{\displaystyle x}
x
1
,
…
,
x
n
{\displaystyle x_{1},\ldots ,\,x_{n}}
(
n
+
m
)
×
(
n
+
m
)
{\displaystyle (n+m)\times (n+m)}
H
¯
(
λ
¯
,
a
)
:=
(
∂
2
L
∂
λ
1
2
…
∂
2
L
∂
λ
1
∂
λ
m
∂
2
L
∂
λ
1
∂
x
1
…
∂
2
L
∂
λ
1
∂
x
n
⋮
⋱
⋮
⋮
⋮
∂
2
L
∂
λ
m
∂
λ
1
…
∂
2
L
∂
λ
m
2
∂
2
L
∂
λ
m
∂
x
1
…
∂
2
L
∂
λ
m
∂
x
n
∂
2
L
∂
x
1
∂
λ
1
…
∂
2
L
∂
x
1
∂
λ
m
∂
2
L
∂
x
1
2
…
∂
2
L
∂
x
1
∂
x
n
⋮
⋮
⋮
⋱
⋮
∂
2
L
∂
x
n
∂
λ
1
…
∂
2
L
∂
x
n
∂
λ
m
∂
2
L
∂
x
n
∂
x
1
…
∂
2
L
∂
x
n
2
)
(
λ
¯
,
a
)
{\displaystyle {\begin{aligned}\operatorname {\overline {H}} ({\bar {\lambda }},\,a)&:=\left({\begin{array}{cccccc}{\frac {\partial ^{2}L}{\partial \lambda _{1}^{2}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{1}\partial \lambda _{m}}}&{\frac {\partial ^{2}L}{\partial \lambda _{1}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{1}\partial x_{n}}}\\\vdots &\ddots &\vdots &\vdots &&\vdots \\{\frac {\partial ^{2}L}{\partial \lambda _{m}\partial \lambda _{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{m}^{2}}}&{\frac {\partial ^{2}L}{\partial \lambda _{m}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial \lambda _{m}\partial x_{n}}}\\{\frac {\partial ^{2}L}{\partial x_{1}\partial \lambda _{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{1}\partial \lambda _{m}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{n}}}\\\vdots &&\vdots &\vdots &\ddots &\vdots \\{\frac {\partial ^{2}L}{\partial x_{n}\partial \lambda _{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{n}\partial \lambda _{m}}}&{\frac {\partial ^{2}L}{\partial x_{n}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{n}^{2}}}\end{array}}\right)({\bar {\lambda }},\,a)\\\end{aligned}}}
bzw. bereits vereinfacht
H
¯
(
λ
¯
,
a
)
:=
(
0
…
0
−
∂
F
1
∂
x
1
…
−
∂
F
1
∂
x
n
⋮
⋱
⋮
⋮
⋮
0
…
0
−
∂
F
m
∂
x
1
…
−
∂
F
m
∂
x
n
−
∂
F
1
∂
x
1
…
−
∂
F
m
∂
x
1
∂
2
L
∂
x
1
2
…
∂
2
L
∂
x
1
∂
x
n
⋮
⋮
⋮
⋱
⋮
−
∂
F
1
∂
x
n
…
−
∂
F
m
∂
x
n
∂
2
L
∂
x
n
∂
x
1
…
∂
2
L
∂
x
n
2
)
(
λ
¯
,
a
)
{\displaystyle {\begin{aligned}\operatorname {\overline {H}} ({\bar {\lambda }},\,a)&:=\left({\begin{array}{cccccc}0&\ldots &0&-{\frac {\partial F_{1}}{\partial x_{1}}}&\ldots &-{\frac {\partial F_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots &\vdots &&\vdots \\0&\ldots &0&-{\frac {\partial F_{m}}{\partial x_{1}}}&\ldots &-{\frac {\partial F_{m}}{\partial x_{n}}}\\-{\frac {\partial F_{1}}{\partial x_{1}}}&\ldots &-{\frac {\partial F_{m}}{\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{n}}}\\\vdots &&\vdots &\vdots &\ddots &\vdots \\-{\frac {\partial F_{1}}{\partial x_{n}}}&\ldots &-{\frac {\partial F_{m}}{\partial x_{n}}}&{\frac {\partial ^{2}L}{\partial x_{n}\partial x_{1}}}&\ldots &{\frac {\partial ^{2}L}{\partial x_{n}^{2}}}\end{array}}\right)({\bar {\lambda }},\,a)\end{aligned}}}
mit
λ
¯
=
λ
¯
1
,
…
,
λ
¯
m
{\displaystyle {\bar {\lambda }}={\bar {\lambda }}_{1},\ldots ,\,{\bar {\lambda }}_{m}}
Für eine zweidimensionale Funktion mit einer Nebenbedingung hat die geränderte Hesse-Matrix folgende Gestalt.
Sei
L
(
x
1
,
x
2
)
=
f
(
x
1
,
x
2
)
+
λ
g
(
x
1
,
x
2
)
{\displaystyle L(x_{1},x_{2})=f(x_{1},x_{2})+\lambda g(x_{1},x_{2})}
Lagrangefunktion , wobei
f
:
R
2
→
R
,
(
x
1
,
x
2
)
↦
f
(
x
1
,
x
2
)
{\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} ,(x_{1},x_{2})\mapsto f(x_{1},x_{2})}
g
(
x
1
,
x
2
)
=
0
{\displaystyle g(x_{1},x_{2})=0\,}
H
¯
(
x
)
=
(
0
g
x
1
g
x
2
g
x
1
L
x
1
x
1
L
x
1
x
2
g
x
2
L
x
2
x
1
L
x
2
x
2
)
=
(
0
∂
g
∂
x
1
∂
g
∂
x
2
∂
g
∂
x
1
∂
2
L
∂
x
1
2
∂
2
L
∂
x
1
∂
x
2
∂
g
∂
x
2
∂
2
L
∂
x
2
∂
x
1
∂
2
L
∂
x
2
2
)
{\displaystyle \operatorname {\bar {H}} (x)={\begin{pmatrix}0&g_{x1}&g_{x2}\\g_{x1}&L_{x1x1}&L_{x1x2}\\g_{x2}&L_{x2x1}&L_{x2x2}\\\end{pmatrix}}={\begin{pmatrix}0&{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial g}{\partial x_{2}}}\\[1.5ex]{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{2}}}\\[1.5ex]{\frac {\partial g}{\partial x_{2}}}&{\frac {\partial ^{2}L}{\partial x_{2}\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{2}^{2}}}\\\end{pmatrix}}}
Die
0
{\displaystyle 0}
H
¯
11
=
∂
2
L
∂
λ
2
{\displaystyle \operatorname {\bar {H}} _{11}={\frac {\partial ^{2}L}{\partial \lambda ^{2}}}}
Eine stationäre Stelle
x
0
{\displaystyle x_{0}}
f
{\displaystyle f}
g
{\displaystyle g}
lokales Maximum , wenn
det
H
¯
(
x
0
)
>
0
{\displaystyle \det {\bar {H}}(x_{0})>0}
lokales Minimum , wenn
det
H
¯
(
x
0
)
<
0
{\displaystyle \det {\bar {H}}(x_{0})<0}
unentscheidbar , wenn
det
H
¯
(
x
0
)
=
0
{\displaystyle \det {\bar {H}}(x_{0})=0}
![{\displaystyle \operatorname {\bar {H}} (x)={\begin{pmatrix}0&g_{x1}&g_{x2}\\g_{x1}&L_{x1x1}&L_{x1x2}\\g_{x2}&L_{x2x1}&L_{x2x2}\\\end{pmatrix}}={\begin{pmatrix}0&{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial g}{\partial x_{2}}}\\[1.5ex]{\frac {\partial g}{\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{1}^{2}}}&{\frac {\partial ^{2}L}{\partial x_{1}\partial x_{2}}}\\[1.5ex]{\frac {\partial g}{\partial x_{2}}}&{\frac {\partial ^{2}L}{\partial x_{2}\partial x_{1}}}&{\frac {\partial ^{2}L}{\partial x_{2}^{2}}}\\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1874c69490ed8fd9584e41335193a0c2ecddf6b)
