z Wikipédie, slobodnej encyklopédie
Kroneckerov súčin (produkt ) matíc , zvyčajne označovaný znamienkom
⊗
{\textstyle \otimes }
matíc , ktorých výsledkom je jedna bloková matica. Pre rozmery (počet riadkov a stĺpcov) výslednej matice platí
A
m
,
n
⊗
B
p
,
q
=
C
m
p
,
n
q
{\displaystyle \mathbf {A} _{m,n}\otimes \mathbf {B} _{p,q}=\mathbf {C} _{mp,nq}}
Pri tomto súčine sa postupuje tak, že každý prvok z matice
A
{\textstyle \mathbf {A} }
B
{\textstyle \mathbf {B} }
A
⊗
B
=
[
a
1
,
1
B
⋯
a
1
,
n
B
⋮
⋱
⋮
a
m
,
1
B
⋯
a
m
,
n
B
]
{\displaystyle \mathbf {A} \otimes \mathbf {B} ={\begin{bmatrix}a_{1,1}\mathbf {B} &\cdots &a_{1,n}\mathbf {B} \\\vdots &\ddots &\vdots \\a_{m,1}\mathbf {B} &\cdots &a_{m,n}\mathbf {B} \\\end{bmatrix}}}
Majme maticu
A
=
[
1
3
5
7
]
{\textstyle \mathbf {A} ={\begin{bmatrix}\color {red}{1}&\color {red}{3}\\\color {red}{5}&\color {red}{7}\\\end{bmatrix}}}
B
=
[
2
4
6
8
]
{\displaystyle \mathbf {B} ={\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}}
A
⊗
B
=
[
1
3
5
7
]
⊗
[
2
4
6
8
]
=
[
1
×
[
2
4
6
8
]
3
×
[
2
4
6
8
]
5
×
[
2
4
6
8
]
7
×
[
2
4
6
8
]
]
=
[
1
×
2
1
×
4
3
×
2
3
×
4
1
×
6
1
×
8
3
×
6
3
×
8
5
×
2
5
×
4
7
×
2
7
×
4
5
×
6
5
×
8
7
×
6
7
×
8
]
=
[
2
4
6
12
6
8
18
24
10
20
14
28
30
40
42
56
]
{\displaystyle {\begin{aligned}\mathbf {A} \otimes \mathbf {B} &={\begin{bmatrix}\color {red}{1}&\color {red}{3}\\\color {red}{5}&\color {red}{7}\\\end{bmatrix}}\otimes {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}\\&=\left[{\begin{array}{c|c}{\color {red}{1}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}&{\color {red}{3}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}\\\hline {\color {red}{5}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}&{\color {red}{7}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}\\\end{array}}\right]\\&=\left[{\begin{array}{cc|cc}{\color {red}{1}}\times {\color {blue}{2}}&{\color {red}{1}}\times {\color {blue}{4}}&{\color {red}{3}}\times {\color {blue}{2}}&{\color {red}{3}}\times {\color {blue}{4}}\\{\color {red}{1}}\times {\color {blue}{6}}&{\color {red}{1}}\times {\color {blue}{8}}&{\color {red}{3}}\times {\color {blue}{6}}&{\color {red}{3}}\times {\color {blue}{8}}\\\hline {\color {red}{5}}\times {\color {blue}{2}}&{\color {red}{5}}\times {\color {blue}{4}}&{\color {red}{7}}\times {\color {blue}{2}}&{\color {red}{7}}\times {\color {blue}{4}}\\{\color {red}{5}}\times {\color {blue}{6}}&{\color {red}{5}}\times {\color {blue}{8}}&{\color {red}{7}}\times {\color {blue}{6}}&{\color {red}{7}}\times {\color {blue}{8}}\\\end{array}}\right]\\&=\left[{\begin{array}{cc|cc}2&4&6&12\\6&8&18&24\\\hline 10&20&14&28\\30&40&42&56\end{array}}\right]\end{aligned}}}
![{\displaystyle {\begin{aligned}\mathbf {A} \otimes \mathbf {B} &={\begin{bmatrix}\color {red}{1}&\color {red}{3}\\\color {red}{5}&\color {red}{7}\\\end{bmatrix}}\otimes {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}\\&=\left[{\begin{array}{c|c}{\color {red}{1}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}&{\color {red}{3}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}\\\hline {\color {red}{5}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}&{\color {red}{7}}\times {\begin{bmatrix}\color {blue}{2}&\color {blue}{4}\\\color {blue}{6}&\color {blue}{8}\\\end{bmatrix}}\\\end{array}}\right]\\&=\left[{\begin{array}{cc|cc}{\color {red}{1}}\times {\color {blue}{2}}&{\color {red}{1}}\times {\color {blue}{4}}&{\color {red}{3}}\times {\color {blue}{2}}&{\color {red}{3}}\times {\color {blue}{4}}\\{\color {red}{1}}\times {\color {blue}{6}}&{\color {red}{1}}\times {\color {blue}{8}}&{\color {red}{3}}\times {\color {blue}{6}}&{\color {red}{3}}\times {\color {blue}{8}}\\\hline {\color {red}{5}}\times {\color {blue}{2}}&{\color {red}{5}}\times {\color {blue}{4}}&{\color {red}{7}}\times {\color {blue}{2}}&{\color {red}{7}}\times {\color {blue}{4}}\\{\color {red}{5}}\times {\color {blue}{6}}&{\color {red}{5}}\times {\color {blue}{8}}&{\color {red}{7}}\times {\color {blue}{6}}&{\color {red}{7}}\times {\color {blue}{8}}\\\end{array}}\right]\\&=\left[{\begin{array}{cc|cc}2&4&6&12\\6&8&18&24\\\hline 10&20&14&28\\30&40&42&56\end{array}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2eb8401d895193c98379944ba00c146a34a39db)
