# Kroneckerjev produkt

Kroneckerjev produkt (oznaka ${\displaystyle \,\otimes \,}$) je operacija, ki se izvaja na dveh matrikah poljubne velikosti, in daje bločno matriko. Kroneckerjevega produkta se ne sme zamenjevati z običajnim množenjem matrik. Kroneckerjev produkt daje matriko tenzorskega produkta.

Imenuje se po nemškem matematiku in logiku Leopoldu Kroneckerju (1823–1891), čeprav ni dokazov, da ga je prvi uporabljal.

## Definicija

Naj bo ${\displaystyle A\,}$ matrika z razsežnostjo ${\displaystyle m\times n\,}$ in naj bo ${\displaystyle B\,}$ z razsežnostjo ${\displaystyle p\times q\,}$, potem je Kroneckerjev produkt ${\displaystyle A\otimes B\,}$ bločna matrika z razsežnostjo ${\displaystyle mp\times nq\,}$:

${\displaystyle \mathbf {A} \otimes \mathbf {B} ={\begin{bmatrix}a_{11}B&\cdots &a_{1n}B\\\vdots &\ddots &\vdots \\a_{m1}B&\cdots &a_{mn}B\end{bmatrix}}.}$.

Bolj točno je to enako:

${\displaystyle \mathbf {A} \otimes \mathbf {B} ={\begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&\cdots &a_{11}b_{1q}&\cdots &\cdots &a_{1n}b_{11}&a_{1n}b_{12}&\cdots &a_{1n}b_{1q}\\a_{11}b_{21}&a_{11}b_{22}&\cdots &a_{11}b_{2q}&\cdots &\cdots &a_{1n}b_{21}&a_{1n}b_{22}&\cdots &a_{1n}b_{2q}\\\vdots &\vdots &\ddots &\vdots &&&\vdots &\vdots &\ddots &\vdots \\a_{11}b_{p1}&a_{11}b_{p2}&\cdots &a_{11}b_{pq}&\cdots &\cdots &a_{1n}b_{p1}&a_{1n}b_{p2}&\cdots &a_{1n}b_{pq}\\\vdots &\vdots &&\vdots &\ddots &&\vdots &\vdots &&\vdots \\\vdots &\vdots &&\vdots &&\ddots &\vdots &\vdots &&\vdots \\a_{m1}b_{11}&a_{m1}b_{12}&\cdots &a_{m1}b_{1q}&\cdots &\cdots &a_{mn}b_{11}&a_{mn}b_{12}&\cdots &a_{mn}b_{1q}\\a_{m1}b_{21}&a_{m1}b_{22}&\cdots &a_{m1}b_{2q}&\cdots &\cdots &a_{mn}b_{21}&a_{mn}b_{22}&\cdots &a_{mn}b_{2q}\\\vdots &\vdots &\ddots &\vdots &&&\vdots &\vdots &\ddots &\vdots \\a_{m1}b_{p1}&a_{m1}b_{p2}&\cdots &a_{m1}b_{pq}&\cdots &\cdots &a_{mn}b_{p1}&a_{mn}b_{p2}&\cdots &a_{mn}b_{pq}\end{bmatrix}}}$.

Če sta ${\displaystyle A\,}$ in ${\displaystyle B\,}$ linearni transformaciji ${\displaystyle V_{1}\to W_{1}\,}$ in ${\displaystyle V_{2}\to W_{2}\,}$, potem je ${\displaystyle A\otimes B\,}$ tenzorski produkt dveh preslikav ${\displaystyle V_{1}\otimes W_{2}\to W_{1}\otimes W_{2}\,}$.

## Zgledi

${\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}\otimes {\begin{bmatrix}5&6\\7&8\end{bmatrix}}={\begin{bmatrix}1\cdot {\begin{bmatrix}5&6\\7&8\end{bmatrix}}&2\cdot {\begin{bmatrix}5&6\\7&8\end{bmatrix}}\\\\3\cdot {\begin{bmatrix}5&6\\7&8\end{bmatrix}}&4\cdot {\begin{bmatrix}5&6\\7&8\end{bmatrix}}\end{bmatrix}}={\begin{bmatrix}5&6&10&12\\7&8&14&16\\15&18&20&24\\21&24&28&32\end{bmatrix}}}$.
${\displaystyle {\begin{bmatrix}1&3&2\\1&0&0\\1&2&2\end{bmatrix}}\otimes {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}={\begin{bmatrix}1\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}&3\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}&2\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}\\\\1\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}&0\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}&0\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}\\\\1\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}&2\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}&2\cdot {\begin{bmatrix}0&5\\5&0\\1&1\end{bmatrix}}\end{bmatrix}}={\begin{bmatrix}0&5&0&15&0&10\\5&0&15&0&10&0\\1&1&3&3&2&2\\0&5&0&0&0&0\\5&0&0&0&0&0\\1&1&0&0&0&0\\0&5&0&10&0&10\\5&0&10&0&10&0\\1&1&2&2&2&2\end{bmatrix}}}$
${\displaystyle {\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}\otimes {\begin{bmatrix}0&5\\6&7\\\end{bmatrix}}={\begin{bmatrix}1\cdot 0&1\cdot 5&2\cdot 0&2\cdot 5\\1\cdot 6&1\cdot 7&2\cdot 6&2\cdot 7\\3\cdot 0&3\cdot 5&4\cdot 0&4\cdot 5\\3\cdot 6&3\cdot 7&4\cdot 6&4\cdot 7\\\end{bmatrix}}={\begin{bmatrix}0&5&0&10\\6&7&12&14\\0&15&0&20\\18&21&24&28\end{bmatrix}}}$.

## Značilnosti

Kroneckerjev produkt je posebni primer tenzorskega produkta:

• ${\displaystyle \mathbf {A} \otimes (\mathbf {B} +\mathbf {C} )=\mathbf {A} \otimes \mathbf {B} +\mathbf {A} \otimes \mathbf {C} ,}$
• ${\displaystyle (\mathbf {A} +\mathbf {B} )\otimes \mathbf {C} =\mathbf {A} \otimes \mathbf {C} +\mathbf {B} \otimes \mathbf {C} ,}$
• ${\displaystyle (k\mathbf {A} )\otimes \mathbf {B} =\mathbf {A} \otimes (k\mathbf {B} )=k(\mathbf {A} \otimes \mathbf {B} ),}$
• ${\displaystyle (\mathbf {A} \otimes \mathbf {B} )\otimes \mathbf {C} =\mathbf {A} \otimes (\mathbf {B} \otimes \mathbf {C} ),}$.

kjer je

• ${\displaystyle A\,}$ matrika
• ${\displaystyle B\,}$ matrika
• ${\displaystyle C\,}$ matrika
• ${\displaystyle k\,}$ skalar

### Komutativnost

Kroneckerjev produkt ni komutativen. To pomeni da sta matriki ${\displaystyle A\otimes B\,}$ in ${\displaystyle B\otimes A\,}$ različni. To se zapiše kot :${\displaystyle A\otimes B\neq B\otimes A}$. Sta pa obe matriki permutacijsko ekvivalentni. To pomeni, da obstajata dve matriki ${\displaystyle P\,}$ in ${\displaystyle Q\,}$ tako, da je:

${\displaystyle \mathbf {A} \otimes \mathbf {B} =\mathbf {P} \,(\mathbf {B} \otimes \mathbf {A} )\,\mathbf {Q} \,}$.

Č e pa sta matriki ${\displaystyle A\,}$ in ${\displaystyle B\,}$ kvadratni, potem sta ${\displaystyle A\otimes B\,}$ ali pa ${\displaystyle B\otimes A\,}$ permutacijsko podobni, kar pomeni, da je ${\displaystyle P=Q^{T}\,}$.

### Mešani produkt

Če so matrike ${\displaystyle A\,}$, ${\displaystyle B\,}$, ${\displaystyle C\,}$ in ${\displaystyle D\,}$ takšne, da se lahko določi ${\displaystyle AC\,}$ in ${\displaystyle BD\,}$, potem velja tudi:

${\displaystyle (\mathbf {A} \otimes \mathbf {B} )(\mathbf {C} \otimes \mathbf {D} )=\mathbf {AC} \otimes \mathbf {BD} }$.

### Transponiranje

Transponiranje Kroneckerjevega produkta da:

${\displaystyle (A\otimes B)^{T}=A^{T}\otimes B^{T}}$.

### Druge značilnosti

${\displaystyle {\overline {A\otimes B}}={\overline {A}}\otimes {\overline {B}}}$.
${\displaystyle (A\otimes B)^{*}=A^{*}\otimes B^{*}}$
• sled je za kvadratne matrike enaka:
${\displaystyle \mathrm {sl} (A\otimes B)=\mathrm {sl} (A)\cdot \mathrm {sl} (B)}$
${\displaystyle \mathrm {rank} (A\otimes B)=\mathrm {rank} (A)\cdot \mathrm {rank} (B)}$
• če ima matrika ${\displaystyle A\,}$ razsežnost ${\displaystyle n\times n\,}$ in matrika ${\displaystyle B\,}$ razsežnost ${\displaystyle m\times m\,}$, potem za determinanto velja:
${\displaystyle \det(A\otimes B)={\det }^{m}(A)\,{\det }^{n}(B)}$
• če so ${\displaystyle (\lambda _{i})_{i=1..n}\,}$ lastne vrednosti matrike ${\displaystyle A\,}$ in ${\displaystyle (\mu _{j})_{j=1..m}\,}$ lastne vrednosti matrike ${\displaystyle B\,}$, potem so:
${\displaystyle (\lambda _{i}\,\mu _{j})_{i=1..n \atop j=1..m}}$ lastne vrednosti matrike ${\displaystyle A\otimes B}$
• kadar sta matriki ${\displaystyle A\,}$ in ${\displaystyle B\,}$ obrnljivi velja tudi:
${\displaystyle (A\otimes B)^{-1}=A^{-1}\otimes B^{-1}}$
• kadar imajo matrike ${\displaystyle A,B,C\,}$ in ${\displaystyle D\,}$ razsežnosti:
• ${\displaystyle A:m\times n\,}$
• ${\displaystyle B:p\times q\,}$
• ${\displaystyle C:n\times r\,}$
• ${\displaystyle D:q\times s\,}$

in sta matriki ${\displaystyle AC\,}$ in ${\displaystyle BD\,}$ definirani, potem velja[1]

${\displaystyle AC\otimes BD=>(A\otimes B)(C\otimes D)}$

## Sklici

1. Steeb (1991), str. 16.

## Viri

• Steeb, Willi-Hans (1991), Kronecker Product of Matrices and Applications, Mannheim ; Wien ; Zürich: BI-Wiss.Verlag, ISBN 3-411-14811-X