Centrirana matrika

Centrirana matrika je simetrična in idempotentna matrika. Če matriko te vrste množimo z vektorjem, je rezultat enak kot, če bi odšteli srednjo vrednost vektorja od vsake komponente.

Definicija

Centrirana matrika z razsežnostjo $n\times n\,$$n\times n\,$ je matrika, ki je določena z

$C_{n}=I_{n}-{\tfrac {1}{n}}\mathbb {O}$$C_{n}=I_{n}-{\tfrac {1}{n}}\mathbb {O}$

kjer je

• $I_{n}\,$$I_{n}\,$ enotska matrika z velikostjo n
• $\mathbb {O} \,$$\mathbb {O} \,$ matrika $n\times n\,$$n\times n\,$ samih enic.

To lahko zapišemo kot

$C_{n}=I_{n}-{\tfrac {1}{n}}\mathbf {1} \mathbf {1} ^{\top }$$C_{n}=I_{n}-{\tfrac {1}{n}}\mathbf {1} \mathbf {1} ^{\top }$

kjer je

• $\mathbf {1} \,$$\mathbf {1} \,$ stolpični vektor z n enicami
• $\top$$\top$ pomeni transponiranje matrike.

Zgled

$C_{1}={\begin{bmatrix}0\end{bmatrix}}$$C_{1}={\begin{bmatrix}0\end{bmatrix}}$,

$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_{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}=\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]$$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]$

Glej tudi

• seznam vrst matrik