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Centrirana matrika je simetrična in idempotentna matrika . Če se matriko te vrste množi z vektorjem , je rezultat enak kot, če bi se odštelo srednjo vrednost vektorja od vsake komponente.
Centrirana matrika z razsežnostjo
n
×
n
{\displaystyle n\times n\!\,}
C
n
=
I
n
−
1
n
O
,
{\displaystyle C_{n}=I_{n}-{\tfrac {1}{n}}\mathbb {O} \!\,,}
kjer je:
I
n
{\displaystyle I_{n}\!\,}
enotska matrika z velikostjo
n
{\displaystyle n\!\,}
O
{\displaystyle \mathbb {O} \!\,}
n
×
n
{\displaystyle n\times n\!\,}
To se lahko zapiše kot:
C
n
=
I
n
−
1
n
1
1
⊤
,
{\displaystyle C_{n}=I_{n}-{\tfrac {1}{n}}\mathbf {1} \mathbf {1} ^{\top }\!\,,}
kjer je:
C
1
=
[
0
]
,
{\displaystyle C_{1}={\begin{bmatrix}0\end{bmatrix}}\!\,,}
C
2
=
[
1
0
0
1
]
−
1
2
[
1
1
1
1
]
=
[
1
2
−
1
2
−
1
2
1
2
]
,
{\displaystyle C_{2}=\left[{\begin{array}{rrr}1&0\\\\0&1\end{array}}\right]-{\frac {1}{2}}\left[{\begin{array}{rrr}1&1\\\\1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {1}{2}}&-{\frac {1}{2}}\\\\-{\frac {1}{2}}&{\frac {1}{2}}\end{array}}\right]\!\,,}
C
3
=
[
1
0
0
0
1
0
0
0
1
]
−
1
3
[
1
1
1
1
1
1
1
1
1
]
=
[
2
3
−
1
3
−
1
3
−
1
3
2
3
−
1
3
−
1
3
−
1
3
2
3
]
.
{\displaystyle C_{3}=\left[{\begin{array}{rrr}1&0&0\\\\0&1&0\\\\0&0&1\end{array}}\right]-{\frac {1}{3}}\left[{\begin{array}{rrr}1&1&1\\\\1&1&1\\\\1&1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {2}{3}}&-{\frac {1}{3}}&-{\frac {1}{3}}\\\\-{\frac {1}{3}}&{\frac {2}{3}}&-{\frac {1}{3}}\\\\-{\frac {1}{3}}&-{\frac {1}{3}}&{\frac {2}{3}}\end{array}}\right]\!\,.}
![{\displaystyle C_{2}=\left[{\begin{array}{rrr}1&0\\\\0&1\end{array}}\right]-{\frac {1}{2}}\left[{\begin{array}{rrr}1&1\\\\1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {1}{2}}&-{\frac {1}{2}}\\\\-{\frac {1}{2}}&{\frac {1}{2}}\end{array}}\right]\!\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/494b8b286a41e3484cca0ec45e56879f1a2c48c8)
![{\displaystyle C_{3}=\left[{\begin{array}{rrr}1&0&0\\\\0&1&0\\\\0&0&1\end{array}}\right]-{\frac {1}{3}}\left[{\begin{array}{rrr}1&1&1\\\\1&1&1\\\\1&1&1\end{array}}\right]=\left[{\begin{array}{rrr}{\frac {2}{3}}&-{\frac {1}{3}}&-{\frac {1}{3}}\\\\-{\frac {1}{3}}&{\frac {2}{3}}&-{\frac {1}{3}}\\\\-{\frac {1}{3}}&-{\frac {1}{3}}&{\frac {2}{3}}\end{array}}\right]\!\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2d535c572fb6b601613fd490d322bbebcadca2)