Kroneckerjev produkt (oznaka
⊗
{\displaystyle \,\otimes \,}
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
uredi
Naj bo
A
{\displaystyle A\,}
m
×
n
{\displaystyle m\times n\,}
B
{\displaystyle B\,}
p
×
q
{\displaystyle p\times q\,}
A
⊗
B
{\displaystyle A\otimes B\,}
m
p
×
n
q
{\displaystyle mp\times nq\,}
A
⊗
B
=
[
a
11
B
⋯
a
1
n
B
⋮
⋱
⋮
a
m
1
B
⋯
a
m
n
B
]
.
{\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:
A
⊗
B
=
[
a
11
b
11
a
11
b
12
⋯
a
11
b
1
q
⋯
⋯
a
1
n
b
11
a
1
n
b
12
⋯
a
1
n
b
1
q
a
11
b
21
a
11
b
22
⋯
a
11
b
2
q
⋯
⋯
a
1
n
b
21
a
1
n
b
22
⋯
a
1
n
b
2
q
⋮
⋮
⋱
⋮
⋮
⋮
⋱
⋮
a
11
b
p
1
a
11
b
p
2
⋯
a
11
b
p
q
⋯
⋯
a
1
n
b
p
1
a
1
n
b
p
2
⋯
a
1
n
b
p
q
⋮
⋮
⋮
⋱
⋮
⋮
⋮
⋮
⋮
⋮
⋱
⋮
⋮
⋮
a
m
1
b
11
a
m
1
b
12
⋯
a
m
1
b
1
q
⋯
⋯
a
m
n
b
11
a
m
n
b
12
⋯
a
m
n
b
1
q
a
m
1
b
21
a
m
1
b
22
⋯
a
m
1
b
2
q
⋯
⋯
a
m
n
b
21
a
m
n
b
22
⋯
a
m
n
b
2
q
⋮
⋮
⋱
⋮
⋮
⋮
⋱
⋮
a
m
1
b
p
1
a
m
1
b
p
2
⋯
a
m
1
b
p
q
⋯
⋯
a
m
n
b
p
1
a
m
n
b
p
2
⋯
a
m
n
b
p
q
]
{\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
A
{\displaystyle A\,}
B
{\displaystyle B\,}
V
1
→
W
1
{\displaystyle V_{1}\to W_{1}\,}
V
2
→
W
2
{\displaystyle V_{2}\to W_{2}\,}
A
⊗
B
{\displaystyle A\otimes B\,}
tenzorski produkt dveh preslikav
V
1
⊗
W
2
→
W
1
⊗
W
2
{\displaystyle V_{1}\otimes W_{2}\to W_{1}\otimes W_{2}\,}
[
1
2
3
4
]
⊗
[
5
6
7
8
]
=
[
1
⋅
[
5
6
7
8
]
2
⋅
[
5
6
7
8
]
3
⋅
[
5
6
7
8
]
4
⋅
[
5
6
7
8
]
]
=
[
5
6
10
12
7
8
14
16
15
18
20
24
21
24
28
32
]
{\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}}}
[
1
3
2
1
0
0
1
2
2
]
⊗
[
0
5
5
0
1
1
]
=
[
1
⋅
[
0
5
5
0
1
1
]
3
⋅
[
0
5
5
0
1
1
]
2
⋅
[
0
5
5
0
1
1
]
1
⋅
[
0
5
5
0
1
1
]
0
⋅
[
0
5
5
0
1
1
]
0
⋅
[
0
5
5
0
1
1
]
1
⋅
[
0
5
5
0
1
1
]
2
⋅
[
0
5
5
0
1
1
]
2
⋅
[
0
5
5
0
1
1
]
]
=
[
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
]
{\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}}}
[
1
2
3
4
]
⊗
[
0
5
6
7
]
=
[
1
⋅
0
1
⋅
5
2
⋅
0
2
⋅
5
1
⋅
6
1
⋅
7
2
⋅
6
2
⋅
7
3
⋅
0
3
⋅
5
4
⋅
0
4
⋅
5
3
⋅
6
3
⋅
7
4
⋅
6
4
⋅
7
]
=
[
0
5
0
10
6
7
12
14
0
15
0
20
18
21
24
28
]
{\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
uredi
Kroneckerjev produkt je posebni primer tenzorskega produkta :
A
⊗
(
B
+
C
)
=
A
⊗
B
+
A
⊗
C
,
{\displaystyle \mathbf {A} \otimes (\mathbf {B} +\mathbf {C} )=\mathbf {A} \otimes \mathbf {B} +\mathbf {A} \otimes \mathbf {C} ,}
(
A
+
B
)
⊗
C
=
A
⊗
C
+
B
⊗
C
,
{\displaystyle (\mathbf {A} +\mathbf {B} )\otimes \mathbf {C} =\mathbf {A} \otimes \mathbf {C} +\mathbf {B} \otimes \mathbf {C} ,}
(
k
A
)
⊗
B
=
A
⊗
(
k
B
)
=
k
(
A
⊗
B
)
,
{\displaystyle (k\mathbf {A} )\otimes \mathbf {B} =\mathbf {A} \otimes (k\mathbf {B} )=k(\mathbf {A} \otimes \mathbf {B} ),}
(
A
⊗
B
)
⊗
C
=
A
⊗
(
B
⊗
C
)
,
{\displaystyle (\mathbf {A} \otimes \mathbf {B} )\otimes \mathbf {C} =\mathbf {A} \otimes (\mathbf {B} \otimes \mathbf {C} ),}
kjer je
A
{\displaystyle A\,}
B
{\displaystyle B\,}
C
{\displaystyle C\,}
k
{\displaystyle k\,}
skalar Komutativnost
uredi
Kroneckerjev produkt ni komutativen . To pomeni da sta matriki
A
⊗
B
{\displaystyle A\otimes B\,}
B
⊗
A
{\displaystyle B\otimes A\,}
A
⊗
B
≠
B
⊗
A
{\displaystyle A\otimes B\neq B\otimes A}
P
{\displaystyle P\,}
Q
{\displaystyle Q\,}
A
⊗
B
=
P
(
B
⊗
A
)
Q
{\displaystyle \mathbf {A} \otimes \mathbf {B} =\mathbf {P} \,(\mathbf {B} \otimes \mathbf {A} )\,\mathbf {Q} \,}
Č e pa sta matriki
A
{\displaystyle A\,}
B
{\displaystyle B\,}
kvadratni , potem sta
A
⊗
B
{\displaystyle A\otimes B\,}
B
⊗
A
{\displaystyle B\otimes A\,}
podobni , kar pomeni, da je
P
=
Q
T
{\displaystyle P=Q^{T}\,}
Mešani produkt
uredi
Če so matrike
A
{\displaystyle A\,}
B
{\displaystyle B\,}
C
{\displaystyle C\,}
D
{\displaystyle D\,}
A
C
{\displaystyle AC\,}
B
D
{\displaystyle BD\,}
(
A
⊗
B
)
(
C
⊗
D
)
=
A
C
⊗
B
D
{\displaystyle (\mathbf {A} \otimes \mathbf {B} )(\mathbf {C} \otimes \mathbf {D} )=\mathbf {AC} \otimes \mathbf {BD} }
Transponiranje
uredi
Transponiranje Kroneckerjevega produkta da:
(
A
⊗
B
)
T
=
A
T
⊗
B
T
{\displaystyle (A\otimes B)^{T}=A^{T}\otimes B^{T}}
Druge značilnosti
uredi
A
⊗
B
¯
=
A
¯
⊗
B
¯
{\displaystyle {\overline {A\otimes B}}={\overline {A}}\otimes {\overline {B}}}
(
A
⊗
B
)
∗
=
A
∗
⊗
B
∗
{\displaystyle (A\otimes B)^{*}=A^{*}\otimes B^{*}}
sled je za kvadratne matrike enaka:
s
l
(
A
⊗
B
)
=
s
l
(
A
)
⋅
s
l
(
B
)
{\displaystyle \mathrm {sl} (A\otimes B)=\mathrm {sl} (A)\cdot \mathrm {sl} (B)}
r
a
n
k
(
A
⊗
B
)
=
r
a
n
k
(
A
)
⋅
r
a
n
k
(
B
)
{\displaystyle \mathrm {rank} (A\otimes B)=\mathrm {rank} (A)\cdot \mathrm {rank} (B)}
če ima matrika
A
{\displaystyle A\,}
n
×
n
{\displaystyle n\times n\,}
B
{\displaystyle B\,}
m
×
m
{\displaystyle m\times m\,}
determinanto velja:
det
(
A
⊗
B
)
=
det
m
(
A
)
det
n
(
B
)
{\displaystyle \det(A\otimes B)={\det }^{m}(A)\,{\det }^{n}(B)}
če so
(
λ
i
)
i
=
1..
n
{\displaystyle (\lambda _{i})_{i=1..n}\,}
lastne vrednosti matrike
A
{\displaystyle A\,}
(
μ
j
)
j
=
1..
m
{\displaystyle (\mu _{j})_{j=1..m}\,}
B
{\displaystyle B\,}
(
λ
i
μ
j
)
i
=
1..
n
j
=
1..
m
{\displaystyle (\lambda _{i}\,\mu _{j})_{i=1..n \atop j=1..m}}
A
⊗
B
{\displaystyle A\otimes B}
kadar sta matriki
A
{\displaystyle A\,}
B
{\displaystyle B\,}
obrnljivi velja tudi:
(
A
⊗
B
)
−
1
=
A
−
1
⊗
B
−
1
{\displaystyle (A\otimes B)^{-1}=A^{-1}\otimes B^{-1}}
kadar imajo matrike
A
,
B
,
C
{\displaystyle A,B,C\,}
D
{\displaystyle D\,}
A
:
m
×
n
{\displaystyle A:m\times n\,}
B
:
p
×
q
{\displaystyle B:p\times q\,}
C
:
n
×
r
{\displaystyle C:n\times r\,}
D
:
q
×
s
{\displaystyle D:q\times s\,}
in sta matriki
A
C
{\displaystyle AC\,}
B
D
{\displaystyle BD\,}
[1]
A
C
⊗
B
D
=>
(
A
⊗
B
)
(
C
⊗
D
)
{\displaystyle AC\otimes BD=>(A\otimes B)(C\otimes D)}
Zunanje povezave
uredi
