Bločna matrika

Bločna matrika (tudi deljena matrika) je matrika, katere elemente lahko razdelimo na dele (bloke).

Primer

Običajno matriko

$\mathbf {P} ={\begin{bmatrix}1&1&2&2\\1&1&2&2\\3&3&4&4\\3&3&4&4\end{bmatrix}}$$\mathbf {P} ={\begin{bmatrix}1&1&2&2\\1&1&2&2\\3&3&4&4\\3&3&4&4\end{bmatrix}}$

lahko razdelimo na 4 skupine (bloke) $2\times 2\,$$2\times 2\,$

$\mathbf {P} _{11}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},\mathbf {P} _{12}={\begin{bmatrix}2&2\\2&2\end{bmatrix}},\mathbf {P} _{21}={\begin{bmatrix}3&3\\3&3\end{bmatrix}},\mathbf {P} _{22}={\begin{bmatrix}4&4\\4&4\end{bmatrix}}.$$\mathbf {P} _{11}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},\mathbf {P} _{12}={\begin{bmatrix}2&2\\2&2\end{bmatrix}},\mathbf {P} _{21}={\begin{bmatrix}3&3\\3&3\end{bmatrix}},\mathbf {P} _{22}={\begin{bmatrix}4&4\\4&4\end{bmatrix}}.$

Tako razdeljeno matriko lahko pišemo kot

$\mathbf {P} ={\begin{bmatrix}\mathbf {P} _{11}&\mathbf {P} _{12}\\\mathbf {P} _{21}&\mathbf {P} _{22}\end{bmatrix}}.$$\mathbf {P} ={\begin{bmatrix}\mathbf {P} _{11}&\mathbf {P} _{12}\\\mathbf {P} _{21}&\mathbf {P} _{22}\end{bmatrix}}.$

Množenje bločnih matrik

Množenje matrik, ki so razdeljene na bloke, lahko pretvorimo na množenje podmatrik.

Če imamo bločno matriko $A\,$$A\,$ z razsežnostjo $m\times p\,$$m\times p\,$ , ki je razdeljena na $q\,$$q\,$ vrstic in $s\,$$s\,$ stolpcev

$\mathbf {A} ={\begin{bmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\cdots &\mathbf {A} _{1s}\\\mathbf {A} _{21}&\mathbf {A} _{22}&\cdots &\mathbf {A} _{2s}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {A} _{q1}&\mathbf {A} _{q2}&\cdots &\mathbf {A} _{qs}\end{bmatrix}}$$\mathbf {A} ={\begin{bmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\cdots &\mathbf {A} _{1s}\\\mathbf {A} _{21}&\mathbf {A} _{22}&\cdots &\mathbf {A} _{2s}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {A} _{q1}&\mathbf {A} _{q2}&\cdots &\mathbf {A} _{qs}\end{bmatrix}}$

in bločno matriko $B\,$$B\,$ z razsežnostjo $p\times n\,$$p\times n\,$, ki je razdeljena na $s\,$$s\,$ vrstic in $r\,$$r\,$ stolpcev

$\mathbf {B} ={\begin{bmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}&\cdots &\mathbf {B} _{1r}\\\mathbf {B} _{21}&\mathbf {B} _{22}&\cdots &\mathbf {B} _{2r}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {B} _{s1}&\mathbf {B} _{s2}&\cdots &\mathbf {B} _{sr}\end{bmatrix}},$$\mathbf {B} ={\begin{bmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}&\cdots &\mathbf {B} _{1r}\\\mathbf {B} _{21}&\mathbf {B} _{22}&\cdots &\mathbf {B} _{2r}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {B} _{s1}&\mathbf {B} _{s2}&\cdots &\mathbf {B} _{sr}\end{bmatrix}},$

potem nam zmnožek

$C=AB\,$$C=AB\,$

da matriko z razsežnostjo $m\times n\,$$m\times n\,$, ki je razdeljena na $q\,$$q\,$ delov (blokov) v vrsticah in $r\,$$r\,$ delov (blokov) v stolpcih. To je

$\mathbf {C} _{\alpha \beta }=\sum _{\gamma =1}^{s}\mathbf {A} _{\alpha \gamma }\mathbf {B} _{\gamma \beta }.$$\mathbf {C} _{\alpha \beta }=\sum _{\gamma =1}^{s}\mathbf {A} _{\alpha \gamma }\mathbf {B} _{\gamma \beta }.$.

kjer je

• $A_{k}\,$$A_{k}\,$ kvadratna matrika v vrstici $k\,$$k\,$.

Bločna diagonalna matrika

Bločna diagonalna matrikaje kvadratna matrika, ki ima na glavni diagonali kvadratne matrike, na vseh blokih izven glavne diagonale pa so ničelne matrike. Takšna matrika ima obliko

$\mathbf {A} ={\begin{bmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{bmatrix}}$$\mathbf {A} ={\begin{bmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{bmatrix}}$

kjer je

• $A_{k}\,$$A_{k}\,$ kvadratna matrika .

Matrika $A\,$$A\,$ je direktna vsota matrik $A_{1},A_{2},\dots ,A_{n}\,$$A_{1},A_{2},\dots ,A_{n}\,$, ki jo zapišemo tudi kot $A_{1}\oplus A_{2}\oplus \dots \oplus A_{n}\,$$A_{1}\oplus A_{2}\oplus \dots \oplus A_{n}\,$.

Determinanta, sled in obratna matrika

Za determinanto in sled bločne matrike velja

$\operatorname {det} \mathbf {A} =\operatorname {det} \mathbf {A} _{1}\cdot \ldots \cdot \operatorname {det} \mathbf {A} _{n}$$\operatorname {det} \mathbf {A} =\operatorname {det} \mathbf {A} _{1}\cdot \ldots \cdot \operatorname {det} \mathbf {A} _{n}$

$\operatorname {sl} \mathbf {A} =\operatorname {sl} \mathbf {A} _{1}+\cdots +\operatorname {sl} \mathbf {A} _{n}.$$\operatorname {sl} \mathbf {A} =\operatorname {sl} \mathbf {A} _{1}+\cdots +\operatorname {sl} \mathbf {A} _{n}.$.

Obratna matrika bločne matrike je tudi obratna matrika, ki jo sestavljajo obratne matrike vsakega bloka

${\begin{pmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{pmatrix}}^{-1}={\begin{pmatrix}\mathbf {A} _{1}^{-1}&0&\cdots &0\\0&\mathbf {A} _{2}^{-1}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}^{-1}\end{pmatrix}}.$${\begin{pmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{pmatrix}}^{-1}={\begin{pmatrix}\mathbf {A} _{1}^{-1}&0&\cdots &0\\0&\mathbf {A} _{2}^{-1}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}^{-1}\end{pmatrix}}.$

Bločna tridiagonalna matrika

Bločna tridiagonalna matrika ima obliko

$\mathbf {A} ={\begin{bmatrix}\mathbf {B} _{1}&\mathbf {C} _{1}&&&\cdots &&0\\\mathbf {A} _{2}&\mathbf {B} _{2}&\mathbf {C} _{2}&&&&\\&\ddots &\ddots &\ddots &&&\vdots \\&&\mathbf {A} _{k}&\mathbf {B} _{k}&\mathbf {C} _{k}&&\\\vdots &&&\ddots &\ddots &\ddots &\\&&&&\mathbf {A} _{n-1}&\mathbf {B} _{n-1}&\mathbf {C} _{n-1}\\0&&\cdots &&&\mathbf {A} _{n}&\mathbf {B} _{n}\end{bmatrix}}$$\mathbf {A} ={\begin{bmatrix}\mathbf {B} _{1}&\mathbf {C} _{1}&&&\cdots &&0\\\mathbf {A} _{2}&\mathbf {B} _{2}&\mathbf {C} _{2}&&&&\\&\ddots &\ddots &\ddots &&&\vdots \\&&\mathbf {A} _{k}&\mathbf {B} _{k}&\mathbf {C} _{k}&&\\\vdots &&&\ddots &\ddots &\ddots &\\&&&&\mathbf {A} _{n-1}&\mathbf {B} _{n-1}&\mathbf {C} _{n-1}\\0&&\cdots &&&\mathbf {A} _{n}&\mathbf {B} _{n}\end{bmatrix}}$

kjer je

• $A_{k}\,$$A_{k}\,$, $B_{k}\,$$B_{k}\,$, $C_{k}\,$$C_{k}\,$kvadratna podmatrika na glavni diagonali ali spodnji ali zgornji stranski diagonali

Njeno strukturo pa lahko opišemo podobno kot pri tridiagonalni matriki

Bločna Teoplitzova matrika

Bločna Toeplitzova matrika ima podobno kot Toeplitzova matrika bloke, ki se ponavljajo vzdolž glavne diagonale matrike

$\mathbf {A} ={\begin{bmatrix}\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&&\cdots &\mathbf {A} _{(1,n-1)}&\mathbf {A} _{(1,n)}\\\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&&&\mathbf {A} _{(1,n-1)}\\&\ddots &\ddots &\ddots &&&\vdots \\&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&\\\vdots &&&\ddots &\ddots &\ddots &\\\mathbf {A} _{(n-1,1)}&&&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}\\\mathbf {A} _{(n,1)}&\mathbf {A} _{(n-1,1)}&\cdots &&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}\end{bmatrix}}.$$\mathbf {A} ={\begin{bmatrix}\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&&\cdots &\mathbf {A} _{(1,n-1)}&\mathbf {A} _{(1,n)}\\\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&&&\mathbf {A} _{(1,n-1)}\\&\ddots &\ddots &\ddots &&&\vdots \\&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&\\\vdots &&&\ddots &\ddots &\ddots &\\\mathbf {A} _{(n-1,1)}&&&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}\\\mathbf {A} _{(n,1)}&\mathbf {A} _{(n-1,1)}&\cdots &&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}\end{bmatrix}}.$.

Direktna vsota

Direktna vsota (oznaka $\oplus \,$$\oplus \,$) matrike $A\,$$A\,$ z razsežnostjo $m\times n\,$$m\times n\,$ in matrike $B\,$$B\,$ z razsežnostjo $p\times q\,$$p\times q\,$ je določena kot

$\mathbf {A} \oplus \mathbf {B} ={\begin{bmatrix}a_{11}&\cdots &a_{1n}&0&\cdots &0\\\vdots &\cdots &\vdots &\vdots &\cdots &\vdots \\a_{m1}&\cdots &a_{mn}&0&\cdots &0\\0&\cdots &0&b_{11}&\cdots &b_{1q}\\\vdots &\cdots &\vdots &\vdots &\cdots &\vdots \\0&\cdots &0&b_{p1}&\cdots &b_{pq}\end{bmatrix}}.$$\mathbf {A} \oplus \mathbf {B} ={\begin{bmatrix}a_{11}&\cdots &a_{1n}&0&\cdots &0\\\vdots &\cdots &\vdots &\vdots &\cdots &\vdots \\a_{m1}&\cdots &a_{mn}&0&\cdots &0\\0&\cdots &0&b_{11}&\cdots &b_{1q}\\\vdots &\cdots &\vdots &\vdots &\cdots &\vdots \\0&\cdots &0&b_{p1}&\cdots &b_{pq}\end{bmatrix}}.$.

Glej tudi

• seznam vrst matrik

Zunanje povezave

• Obratna matrika bločne matrike v Priročniku za matrike (v angleščini)
• Determinanta bločne matrike v Priročniku za matrike (v angleščini)
• Bločna matrika na MathWorld (v angleščini)