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 A {\displaystyle A\,} matrika z razsežnostjo m × n {\displaystyle m\times n\,} in naj bo B {\displaystyle B\,} z razsežnostjo p × q {\displaystyle p\times q\,} , potem je Kroneckerjev produkt A ⊗ B {\displaystyle A\otimes B\,} bločna matrika z razsežnostjo 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\,} in B {\displaystyle B\,} linearni transformaciji V 1 → W 1 {\displaystyle V_{1}\to W_{1}\,} in V 2 → W 2 {\displaystyle V_{2}\to W_{2}\,} , potem je 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}\,} .
Zgledi [ 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 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\,} matrika B {\displaystyle B\,} matrika C {\displaystyle C\,} matrika k {\displaystyle k\,} skalar
Komutativnost Kroneckerjev produkt ni komutativen . To pomeni da sta matriki A ⊗ B {\displaystyle A\otimes B\,} in B ⊗ A {\displaystyle B\otimes A\,} različni. To se zapiše kot : A ⊗ B ≠ B ⊗ A {\displaystyle A\otimes B\neq B\otimes A} . Sta pa obe matriki permutacijsko ekvivalentni. To pomeni, da obstajata dve matriki P {\displaystyle P\,} in Q {\displaystyle Q\,} tako, da je:
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\,} in B {\displaystyle B\,} kvadratni , potem sta A ⊗ B {\displaystyle A\otimes B\,} ali pa B ⊗ A {\displaystyle B\otimes A\,} permutacijsko podobni , kar pomeni, da je P = Q T {\displaystyle P=Q^{T}\,} .
Mešani produkt Če so matrike A {\displaystyle A\,} , B {\displaystyle B\,} , C {\displaystyle C\,} in D {\displaystyle D\,} takšne, da se lahko določi A C {\displaystyle AC\,} in B D {\displaystyle BD\,} , potem velja tudi:
( 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 Transponiranje Kroneckerjevega produkta da:
( A ⊗ B ) T = A T ⊗ B T {\displaystyle (A\otimes B)^{T}=A^{T}\otimes B^{T}} .
Druge značilnosti 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\,} razsežnost n × n {\displaystyle n\times n\,} in matrika B {\displaystyle B\,} razsežnost m × m {\displaystyle m\times m\,} , potem za 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\,} in ( μ j ) j = 1.. m {\displaystyle (\mu _{j})_{j=1..m}\,} lastne vrednosti matrike B {\displaystyle B\,} , potem so: ( λ i μ j ) i = 1.. n j = 1.. m {\displaystyle (\lambda _{i}\,\mu _{j})_{i=1..n \atop j=1..m}} lastne vrednosti matrike A ⊗ B {\displaystyle A\otimes B} kadar sta matriki A {\displaystyle A\,} in 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\,} in D {\displaystyle D\,} razsežnosti: 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\,} in B D {\displaystyle BD\,} definirani, potem velja[1]
A C ⊗ B D => ( A ⊗ B ) ( C ⊗ D ) {\displaystyle AC\otimes BD=>(A\otimes B)(C\otimes D)}
Sklici
Viri Steeb, Willi-Hans (1991), Kronecker Product of Matrices and Applications , Mannheim ; Wien ; Zürich: BI-Wiss.Verlag, ISBN 3-411-14811-X
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