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Arrowhead-like matrix
In mathematics , an arrowhead-like matrix is a matrix of the form
[
X
m
×
m
B
o
,
m
×
n
0
m
×
k
C
o
,
n
×
m
D
n
×
n
C
v
,
n
×
k
0
k
×
m
B
v
,
k
×
n
0
k
×
k
]
{\displaystyle {\begin{bmatrix}X_{m\times m}&B_{o,m\times n}&0_{m\times k}\\C_{o,n\times m}&D_{n\times n}&C_{v,n\times k}\\0_{k\times m}&B_{v,k\times n}&0_{k\times k}\end{bmatrix}}}
where X is a diagonal matrix.
When n ≥ k , the arrowhead-like matrix has its inverse matrix
[
X
m
×
m
B
o
,
m
×
n
0
m
×
k
C
o
,
n
×
m
D
n
×
n
C
v
,
n
×
k
0
k
×
m
B
v
,
k
×
n
0
k
×
k
]
−
1
=
[
X
−
1
+
X
−
1
B
o
(
D
−
D
C
v
[
B
v
D
C
v
]
−
1
B
v
D
)
C
o
X
−
1
−
X
−
1
B
o
(
D
−
D
C
v
[
B
v
D
C
v
]
−
1
B
v
D
)
−
X
−
1
B
o
D
C
v
[
B
v
D
C
v
]
−
1
−
(
D
−
D
C
v
[
B
v
D
C
v
]
−
1
B
v
D
)
C
o
X
−
1
D
−
D
C
v
[
B
v
D
C
v
]
−
1
B
v
D
D
C
v
[
B
v
D
C
v
]
−
1
−
[
B
v
D
C
v
]
−
1
B
v
D
C
o
X
−
1
[
B
v
D
C
v
]
−
1
B
v
D
−
[
B
v
D
C
v
]
−
1
]
{\displaystyle {\begin{bmatrix}X_{m\times m}&B_{o,m\times n}&0_{m\times k}\\C_{o,n\times m}&D_{n\times n}&C_{v,n\times k}\\0_{k\times m}&B_{v,k\times n}&0_{k\times k}\end{bmatrix}}^{-1}={\begin{bmatrix}X^{-1}+X^{-1}B_{o}({\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}})C_{o}X^{-1}&-X^{-1}B_{o}({\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}})&-X^{-1}B_{o}{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}\\-({\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}})C_{o}X^{-1}&{\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}}&{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}\\-[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}}C_{o}X^{-1}&[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}}&-[B_{v}{\mathcal {D}}C_{v}]^{-1}\end{bmatrix}}}
D
=
(
D
−
C
o
X
−
1
B
o
)
−
1
{\displaystyle {\mathcal {D}}=(D-C_{o}X^{-1}B_{o})^{-1}}
special cases:
k = 0:
[
X
B
o
C
o
D
]
−
1
=
[
X
−
1
+
X
−
1
B
o
D
C
o
X
−
1
−
X
−
1
B
o
D
−
D
C
o
X
−
1
D
]
{\displaystyle {\begin{bmatrix}X&B_{o}\\C_{o}&D\end{bmatrix}}^{-1}={\begin{bmatrix}X^{-1}+X^{-1}B_{o}{\mathcal {D}}C_{o}X^{-1}&-X^{-1}B_{o}{\mathcal {D}}\\-{\mathcal {D}}C_{o}X^{-1}&{\mathcal {D}}\end{bmatrix}}}
k = n :
[
X
B
o
0
C
o
D
C
v
0
B
v
0
]
−
1
=
[
X
−
1
0
−
X
−
1
B
o
B
v
−
1
0
0
B
v
−
1
−
C
v
−
1
C
o
X
−
1
C
v
−
1
−
C
v
−
1
(
D
−
C
o
X
−
1
B
o
)
B
v
−
1
]
{\displaystyle {\begin{bmatrix}X&B_{o}&0\\C_{o}&D&C_{v}\\0&B_{v}&0\end{bmatrix}}^{-1}={\begin{bmatrix}X^{-1}&0&-X^{-1}B_{o}B_{v}^{-1}\\0&0&B_{v}^{-1}\\-C_{v}^{-1}C_{o}X^{-1}&C_{v}^{-1}&-C_{v}^{-1}(D-C_{o}X^{-1}B_{o})B_{v}^{-1}\end{bmatrix}}}
![{\displaystyle {\begin{bmatrix}X_{m\times m}&B_{o,m\times n}&0_{m\times k}\\C_{o,n\times m}&D_{n\times n}&C_{v,n\times k}\\0_{k\times m}&B_{v,k\times n}&0_{k\times k}\end{bmatrix}}^{-1}={\begin{bmatrix}X^{-1}+X^{-1}B_{o}({\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}})C_{o}X^{-1}&-X^{-1}B_{o}({\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}})&-X^{-1}B_{o}{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}\\-({\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}})C_{o}X^{-1}&{\mathcal {D}}-{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}}&{\mathcal {D}}C_{v}[B_{v}{\mathcal {D}}C_{v}]^{-1}\\-[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}}C_{o}X^{-1}&[B_{v}{\mathcal {D}}C_{v}]^{-1}B_{v}{\mathcal {D}}&-[B_{v}{\mathcal {D}}C_{v}]^{-1}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/002f14148237f2474602e900471913328e6ff11a)
