Matrix Representation of Tensor Product
📂Linear Algebra Matrix Representation of Tensor Product Buildup Choose bases V , V ′ \mathcal{V}, {\mathcal{V}}^{\prime} V , V ′ finite-dimensional vector spaces V , V ′ V, V^{\prime} V , V ′ linear transformation ϕ : V → V ′ \phi : V \to V^{\prime} ϕ : V → V ′ matrix representation ϕ \phi ϕ V , V ′ , W , W ′ V, V^{\prime}, W, W^{\prime} V , V ′ , W , W ′ ordered basis V , V ′ , W , W ′ \mathcal{V}, {\mathcal{V}}^{\prime}, \mathcal{W}, {\mathcal{W}}^{\prime} V , V ′ , W , W ′ ϕ : V → V ′ \phi : V \to V^{\prime} ϕ : V → V ′ ψ : W → W ′ \psi : W \to W^{\prime} ψ : W → W ′
n = dim V , m = dim V ′ , p = dim W , q = dim W ′
n = \dim V,\quad m = \dim V^{\prime},\quad p = \dim W,\quad q = \dim W^{\prime}
n = dim V , m = dim V ′ , p = dim W , q = dim W ′
V = { v i } i = 1 n , V ′ = { v j ′ } j = 1 m , W = { w k } k = 1 p , W ′ = { w l ′ } l = 1 q
\mathcal{V} = \left\{ v_{i} \right\}_{i=1}^{n},\quad {\mathcal{V}}^{\prime} = \left\{ v_{j}^{\prime} \right\}_{j=1}^{m},\quad \mathcal{W} = \left\{ w_{k} \right\}_{k=1}^{p},\quad {\mathcal{W}}^{\prime} = \left\{ w_{l}^{\prime} \right\}_{l=1}^{q}
V = { v i } i = 1 n , V ′ = { v j ′ } j = 1 m , W = { w k } k = 1 p , W ′ = { w l ′ } l = 1 q
There exist matrix representations for the two linear transformations ϕ \phi ϕ ψ \psi ψ
A = [ ϕ ] V V ′ ∈ M m × n B = [ ψ ] W W ′ ∈ M q × p
A = \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}}\in M_{m \times n} \qquad B = \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}} \in M_{q \times p}
A = [ ϕ ] V V ′ ∈ M m × n B = [ ψ ] W W ′ ∈ M q × p
Notate the ordered basis of the tensor product V ⊗ W V \otimes W V ⊗ W V ⊗ W = { v i ⊗ w k } \mathcal{V} \otimes \mathcal{W} = \left\{ v_{i} \otimes w_{k} \right\} V ⊗ W = { v i ⊗ w k }
v 1 ⊗ w 1 , … , v 1 × w p , v 2 ⊗ w 1 , … , v 2 × w p , … v n ⊗ w 1 , … , v n × w p
v_{1} \otimes w_{1}, \dots, v_{1} \times w_{p}, \\
v_{2}\otimes w_{1}, \dots, v_{2} \times w_{p}, \\
\dots \\
v_{n}\otimes w_{1}, \dots, v_{n} \times w_{p}
v 1 ⊗ w 1 , … , v 1 × w p , v 2 ⊗ w 1 , … , v 2 × w p , … v n ⊗ w 1 , … , v n × w p
Let’s give an order to the basis V ′ ⊗ W ′ = { v j ′ ⊗ w l ′ } \mathcal{V}^{\prime} \otimes \mathcal{W}^{\prime} = \left\{ v_{j}^{\prime} \otimes w_{l}^{\prime} \right\} V ′ ⊗ W ′ = { v j ′ ⊗ w l ′ } V ′ ⊗ W ′ V^{\prime} \otimes W^{\prime} V ′ ⊗ W ′ ϕ \phi ϕ ψ \psi ψ ϕ ⊗ ψ : V ⊗ W → V ′ ⊗ W ′ \phi \otimes \psi : V \otimes W \to V^{\prime} \otimes W^{\prime} ϕ ⊗ ψ : V ⊗ W → V ′ ⊗ W ′ matrix representation as shown below.
[ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′
\begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}}
[ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′
Theorem Let the matrix representations of the two linear transformations ϕ : V → V ′ \phi : V \to V^{\prime} ϕ : V → V ′ ψ : W → W ′ \psi : W \to W^{\prime} ψ : W → W ′ A = [ ϕ ] V V ′ A = \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}} A = [ ϕ ] V V ′ B = [ ψ ] W W ′ B = \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}} B = [ ψ ] W W ′ ϕ ⊗ ψ : V ⊗ W → V ′ ⊗ W ′ \phi \otimes \psi : V \otimes W \to V^{\prime} \otimes W^{\prime} ϕ ⊗ ψ : V ⊗ W → V ′ ⊗ W ′ Kronecker product of A A A B B B
[ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′ = A ⊗ B = [ ϕ ] V V ′ ⊗ [ ψ ] W W ′
\begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} = A \otimes B = \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}} \otimes \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}}
[ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′ = A ⊗ B = [ ϕ ] V V ′ ⊗ [ ψ ] W W ′
Proof To find the matrix representation [ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′ \begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} [ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′ we must see how the basis V ⊗ W \mathcal{V} \otimes \mathcal{W} V ⊗ W ϕ ⊗ ψ \phi \otimes \psi ϕ ⊗ ψ Suppose the matrix representations of the two linear transformations are as follows.
[ ϕ ] V V ′ = A = [ α j i ] ∈ M m × n [ ψ ] W W ′ = B = [ β l k ] ∈ M q × p
\begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}} = A = [ \alpha_{ji} ] \in M_{m \times n} \qquad \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}} = B = [ \beta_{lk} ] \in M_{q \times p}
[ ϕ ] V V ′ = A = [ α ji ] ∈ M m × n [ ψ ] W W ′ = B = [ β l k ] ∈ M q × p
Which means ϕ ( v i ) = ∑ j α j i v j ′ \phi (v_{i}) = \sum\limits_{j}\alpha_{ji}v_{j}^{\prime} ϕ ( v i ) = j ∑ α ji v j ′ ψ ( w k ) = ∑ l β l k w k ′ \psi (w_{k}) = \sum\limits_{l}\beta_{lk}w_{k}^{\prime} ψ ( w k ) = l ∑ β l k w k ′ tensor product of linear transformations and product vector , the basis vector v i ⊗ w k v_{i} \otimes w_{k} v i ⊗ w k
( ϕ ⊗ ψ ) ( v i ⊗ w k ) = ϕ ( v i ) ⊗ ψ ( w k ) = ( ∑ j α j i v j ′ ) ⊗ ( ∑ l β l k w l ′ ) = ∑ j , l α j i β l k v j ′ ⊗ w l ′
\begin{align*}
(\phi \otimes \psi)(v_{i} \otimes w_{k})
&= \phi (v_{i}) \otimes \psi (w_{k}) \\
&= \left( \sum\limits_{j}\alpha_{ji}v_{j}^{\prime} \right) \otimes \left( \sum\limits_{l}\beta_{lk}w_{l}^{\prime} \right) \\
&= \sum_{j,l} \alpha_{ji}\beta_{lk} v_{j}^{\prime} \otimes w_{l}^{\prime} \\
\end{align*}
( ϕ ⊗ ψ ) ( v i ⊗ w k ) = ϕ ( v i ) ⊗ ψ ( w k ) = ( j ∑ α ji v j ′ ) ⊗ ( l ∑ β l k w l ′ ) = j , l ∑ α ji β l k v j ′ ⊗ w l ′
⟹ [ ( ϕ ⊗ ψ ) ( v i ⊗ w k ) ] V ′ ⊗ W ′ = [ α 1 i β 1 k α 1 i β 2 k ⋮ α 1 i β q k α 2 i β 1 k α 2 i β 2 k ⋮ α 2 i β q k ⋮ α m i β 1 k α m i β 2 k ⋮ α m i β q k ]
\implies \left[ (\phi \otimes \psi)(v_{i} \otimes w_{k}) \right]_{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} = \begin{bmatrix}
\alpha_{1i}\beta_{1k} \\
\alpha_{1i}\beta_{2k} \\
\vdots \\
\alpha_{1i}\beta_{qk} \\
\alpha_{2i}\beta_{1k} \\
\alpha_{2i}\beta_{2k} \\
\vdots \\
\alpha_{2i}\beta_{qk} \\
\vdots \\
\alpha_{mi}\beta_{1k} \\
\alpha_{mi}\beta_{2k} \\
\vdots \\
\alpha_{mi}\beta_{qk} \\
\end{bmatrix}
⟹ [ ( ϕ ⊗ ψ ) ( v i ⊗ w k ) ] V ′ ⊗ W ′ = α 1 i β 1 k α 1 i β 2 k ⋮ α 1 i β q k α 2 i β 1 k α 2 i β 2 k ⋮ α 2 i β q k ⋮ α mi β 1 k α mi β 2 k ⋮ α mi β q k
Thus, summarizing we get the following.
[ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′ = [ [ ϕ ( v 1 ) ⊗ ψ ( w 1 ) ] V ′ ⊗ W ′ [ ϕ ( v 1 ) ⊗ ψ ( w 2 ) ] V ′ ⊗ W ′ ⋯ [ ϕ ( v n ) ⊗ ψ ( w p ) ] V ′ ⊗ W ′ ] = [ α 11 β 11 α 11 β 12 ⋯ α 11 β 1 p ⋯ α 1 n β 11 α 1 n β 12 ⋯ α 1 n β 1 p α 11 β 21 α 11 β 22 ⋯ α 11 β 2 p ⋯ α 1 n β 21 α 1 n β 22 ⋯ α 1 n β 2 p ⋮ ⋮ ⋱ ⋮ ⋯ ⋮ ⋮ ⋱ ⋮ α 11 β q 1 α 11 β q 2 ⋯ α 11 β q p ⋯ α 1 n β q 1 α 1 n β q 2 ⋯ α 1 n β q p α 21 β 11 α 21 β 12 ⋯ α 21 β 1 p ⋯ α 2 n β 11 α 2 n β 12 ⋯ α 2 n β 1 p α 21 β 21 α 21 β 22 ⋯ α 21 β 2 p ⋯ α 2 n β 21 α 2 n β 22 ⋯ α 2 n β 2 p ⋮ ⋮ ⋱ ⋮ ⋯ ⋮ ⋮ ⋱ ⋮ α 21 β q 1 α 21 β q 2 ⋯ α 21 β q p ⋯ α 2 n β q 1 α 2 n β q 2 ⋯ α 2 n β q p ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ α m 1 β 11 α m 1 β 12 ⋯ α m 1 β 1 p ⋯ α m n β 11 α m n β 12 ⋯ α m n β 1 p α m 1 β 21 α m 1 β 22 ⋯ α m 1 β 2 p ⋯ α m n β 21 α m n β 22 ⋯ α m n β 2 p ⋮ ⋮ ⋱ ⋮ ⋯ ⋮ ⋮ ⋱ ⋮ α m 1 β q 1 α m 1 β q 2 ⋯ α m 1 β q p ⋯ α m n β q 1 α m n β q 2 ⋯ α m n β q p ] = [ α 11 [ β 11 β 12 ⋯ β 1 p β 21 β 22 ⋯ β 2 p ⋮ ⋮ ⋱ ⋮ β q 1 β q 2 ⋯ β q p ] ⋯ α 1 n [ β 11 β 12 ⋯ β 1 p β 21 β 22 ⋯ β 2 p ⋮ ⋮ ⋱ ⋮ β q 1 β q 2 ⋯ β q p ] α 21 [ β 11 β 12 ⋯ β 1 p β 21 β 22 ⋯ β 2 p ⋮ ⋮ ⋱ ⋮ β q 1 β q 2 ⋯ β q p ] ⋯ α 2 n [ β 11 β 12 ⋯ β 1 p β 21 β 22 ⋯ β 2 p ⋮ ⋮ ⋱ ⋮ β q 1 β q 2 ⋯ β q p ] ⋮ ⋱ ⋮ α m 1 [ β 11 β 12 ⋯ β 1 p β 21 β 22 ⋯ β 2 p ⋮ ⋮ ⋱ ⋮ β q 1 β q 2 ⋯ β q p ] ⋯ α m n [ β 11 β 12 ⋯ β 1 p β 21 β 22 ⋯ β 2 p ⋮ ⋮ ⋱ ⋮ β q 1 β q 2 ⋯ β q p ] ] = [ α 11 B ⋯ α 1 n B α 21 B ⋯ α 2 n B ⋮ ⋱ ⋮ α m 1 B ⋯ α m n B ] = A ⊗ B
\begin{align*}
& \begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} \\
&= \begin{bmatrix}
\left[ \phi (v_{1}) \otimes \psi (w_{1}) \right]_{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} &
\left[ \phi (v_{1}) \otimes \psi (w_{2}) \right]_{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} &
\cdots &
\left[ \phi (v_{n}) \otimes \psi (w_{p}) \right]_{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}}
\end{bmatrix} \\
&=
\left[
\begin{array}{cccc|c|cccc}
\alpha_{11}\beta_{11} & \alpha_{11}\beta_{12} & \cdots & \alpha_{11}\beta_{1p} & \cdots & \alpha_{1n}\beta_{11} & \alpha_{1n}\beta_{12} & \cdots & \alpha_{1n}\beta_{1p} & \\
\alpha_{11}\beta_{21} & \alpha_{11}\beta_{22} & \cdots & \alpha_{11}\beta_{2p} & \cdots & \alpha_{1n}\beta_{21} & \alpha_{1n}\beta_{22} & \cdots & \alpha_{1n}\beta_{2p} & \\
\vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \vdots & \ddots & \vdots \\
\alpha_{11}\beta_{q1} & \alpha_{11}\beta_{q2} & \cdots & \alpha_{11}\beta_{qp} & \cdots & \alpha_{1n}\beta_{q1} & \alpha_{1n}\beta_{q2} & \cdots & \alpha_{1n}\beta_{qp} & \\ \hline
\alpha_{21}\beta_{11} & \alpha_{21}\beta_{12} & \cdots & \alpha_{21}\beta_{1p} & \cdots & \alpha_{2n}\beta_{11} & \alpha_{2n}\beta_{12} & \cdots & \alpha_{2n}\beta_{1p} & \\
\alpha_{21}\beta_{21} & \alpha_{21}\beta_{22} & \cdots & \alpha_{21}\beta_{2p} & \cdots & \alpha_{2n}\beta_{21} & \alpha_{2n}\beta_{22} & \cdots & \alpha_{2n}\beta_{2p} & \\
\vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \vdots & \ddots & \vdots \\
\alpha_{21}\beta_{q1} & \alpha_{21}\beta_{q2} & \cdots & \alpha_{21}\beta_{qp} & \cdots & \alpha_{2n}\beta_{q1} & \alpha_{2n}\beta_{q2} & \cdots & \alpha_{2n}\beta_{qp} & \\ \hline
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ \hline
\alpha_{m1}\beta_{11} & \alpha_{m1}\beta_{12} & \cdots & \alpha_{m1}\beta_{1p} & \cdots & \alpha_{mn}\beta_{11} & \alpha_{mn}\beta_{12} & \cdots & \alpha_{mn}\beta_{1p} & \\
\alpha_{m1}\beta_{21} & \alpha_{m1}\beta_{22} & \cdots & \alpha_{m1}\beta_{2p} & \cdots & \alpha_{mn}\beta_{21} & \alpha_{mn}\beta_{22} & \cdots & \alpha_{mn}\beta_{2p} & \\
\vdots & \vdots & \ddots & \vdots & \cdots & \vdots & \vdots & \ddots & \vdots \\
\alpha_{m1}\beta_{q1} & \alpha_{m1}\beta_{q2} & \cdots & \alpha_{m1}\beta_{qp} & \cdots & \alpha_{mn}\beta_{q1} & \alpha_{mn}\beta_{q2} & \cdots & \alpha_{mn}\beta_{qp} & \\
\end{array}
\right] \\
&=
\left[
\begin{array}{c}
\alpha_{11} \begin{bmatrix}
\beta_{11} & \beta_{12} & \cdots & \beta_{1p} \\
\beta_{21} & \beta_{22} & \cdots & \beta_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\beta_{q1} & \beta_{q2} & \cdots & \beta_{qp}
\end{bmatrix}
& \cdots &
\alpha_{1n}\begin{bmatrix}
\beta_{11} & \beta_{12} & \cdots & \beta_{1p} \\
\beta_{21} & \beta_{22} & \cdots & \beta_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\beta_{q1} & \beta_{q2} & \cdots & \beta_{qp}
\end{bmatrix} \\
\alpha_{21} \begin{bmatrix}
\beta_{11} & \beta_{12} & \cdots & \beta_{1p} \\
\beta_{21} & \beta_{22} & \cdots & \beta_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\beta_{q1} & \beta_{q2} & \cdots & \beta_{qp}
\end{bmatrix}
& \cdots &
\alpha_{2n}\begin{bmatrix}
\beta_{11} & \beta_{12} & \cdots & \beta_{1p} \\
\beta_{21} & \beta_{22} & \cdots & \beta_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\beta_{q1} & \beta_{q2} & \cdots & \beta_{qp}
\end{bmatrix} \\
\vdots & \ddots & \vdots \\
\alpha_{m1} \begin{bmatrix}
\beta_{11} & \beta_{12} & \cdots & \beta_{1p} \\
\beta_{21} & \beta_{22} & \cdots & \beta_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\beta_{q1} & \beta_{q2} & \cdots & \beta_{qp}
\end{bmatrix}
& \cdots &
\alpha_{mn}\begin{bmatrix}
\beta_{11} & \beta_{12} & \cdots & \beta_{1p} \\
\beta_{21} & \beta_{22} & \cdots & \beta_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\beta_{q1} & \beta_{q2} & \cdots & \beta_{qp}
\end{bmatrix} \\
\end{array}
\right] \\
&=
\begin{bmatrix}
\alpha_{11} B & \cdots & \alpha_{1n} B \\
\alpha_{21} B & \cdots & \alpha_{2n} B \\
\vdots & \ddots & \vdots \\
\alpha_{m1} B & \cdots & \alpha_{mn} B
\end{bmatrix} \\
&= A \otimes B
\end{align*}
[ ϕ ⊗ ψ ] V ⊗ W V ′ ⊗ W ′ = [ [ ϕ ( v 1 ) ⊗ ψ ( w 1 ) ] V ′ ⊗ W ′ [ ϕ ( v 1 ) ⊗ ψ ( w 2 ) ] V ′ ⊗ W ′ ⋯ [ ϕ ( v n ) ⊗ ψ ( w p ) ] V ′ ⊗ W ′ ] = α 11 β 11 α 11 β 21 ⋮ α 11 β q 1 α 21 β 11 α 21 β 21 ⋮ α 21 β q 1 ⋮ α m 1 β 11 α m 1 β 21 ⋮ α m 1 β q 1 α 11 β 12 α 11 β 22 ⋮ α 11 β q 2 α 21 β 12 α 21 β 22 ⋮ α 21 β q 2 ⋮ α m 1 β 12 α m 1 β 22 ⋮ α m 1 β q 2 ⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋱ ⋯ ⋮ ⋯ ⋯ ⋱ ⋯ α 11 β 1 p α 11 β 2 p ⋮ α 11 β qp α 21 β 1 p α 21 β 2 p ⋮ α 21 β qp ⋮ α m 1 β 1 p α m 1 β 2 p ⋮ α m 1 β qp ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋯ α 1 n β 11 α 1 n β 21 ⋮ α 1 n β q 1 α 2 n β 11 α 2 n β 21 ⋮ α 2 n β q 1 ⋮ α mn β 11 α mn β 21 ⋮ α mn β q 1 α 1 n β 12 α 1 n β 22 ⋮ α 1 n β q 2 α 2 n β 12 α 2 n β 22 ⋮ α 2 n β q 2 ⋮ α mn β 12 α mn β 22 ⋮ α mn β q 2 ⋯ ⋯ ⋱ ⋯ ⋯ ⋯ ⋱ ⋯ ⋮ ⋯ ⋯ ⋱ ⋯ α 1 n β 1 p α 1 n β 2 p ⋮ α 1 n β qp α 2 n β 1 p α 2 n β 2 p ⋮ α 2 n β qp ⋮ α mn β 1 p α mn β 2 p ⋮ α mn β qp = α 11 β 11 β 21 ⋮ β q 1 β 12 β 22 ⋮ β q 2 ⋯ ⋯ ⋱ ⋯ β 1 p β 2 p ⋮ β qp α 21 β 11 β 21 ⋮ β q 1 β 12 β 22 ⋮ β q 2 ⋯ ⋯ ⋱ ⋯ β 1 p β 2 p ⋮ β qp ⋮ α m 1 β 11 β 21 ⋮ β q 1 β 12 β 22 ⋮ β q 2 ⋯ ⋯ ⋱ ⋯ β 1 p β 2 p ⋮ β qp ⋯ ⋯ ⋱ ⋯ α 1 n β 11 β 21 ⋮ β q 1 β 12 β 22 ⋮ β q 2 ⋯ ⋯ ⋱ ⋯ β 1 p β 2 p ⋮ β qp α 2 n β 11 β 21 ⋮ β q 1 β 12 β 22 ⋮ β q 2 ⋯ ⋯ ⋱ ⋯ β 1 p β 2 p ⋮ β qp ⋮ α mn β 11 β 21 ⋮ β q 1 β 12 β 22 ⋮ β q 2 ⋯ ⋯ ⋱ ⋯ β 1 p β 2 p ⋮ β qp = α 11 B α 21 B ⋮ α m 1 B ⋯ ⋯ ⋱ ⋯ α 1 n B α 2 n B ⋮ α mn B = A ⊗ B
■
