텐서곱의 행렬표현 📂선형대수

텐서곱의 행렬표현

Matrix Representation for Tensor Product

빌드업1

유한차원 벡터공간 $V, V^{\prime}$에 대해서 각각 기저저 $\mathcal{V}, {\mathcal{V}}^{\prime}$를 선택하자. 그러면 선형변환 $\phi : V \to V^{\prime}$과 동치인 행렬 $\begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}}$이 존재하고, 이를 $\phi$의 행렬표현이라 한다. 이제 유한차원 벡터공간 $V, V^{\prime}, W, W^{\prime}$와 이들의 순서기저 $\mathcal{V}, {\mathcal{V}}^{\prime}, \mathcal{W}, {\mathcal{W}}^{\prime}$, 그리고 두 선형변환 $\phi : V \to V^{\prime}$, $\psi : W \to W^{\prime}$가 주어졌다고 하자.

$$ n = \dim V,\quad m = \dim V^{\prime},\quad p = \dim W,\quad q = \dim W^{\prime} $$

$$ \mathcal{V} = \left\{ v_{i} \right\}_{i=1}^{n},\quad {\mathcal{V}}^{\prime} = \left\{ v_{j}^{\prime} \right\}_{j=1}^{m},\quad \mathcal{W} = \left\{ w_{k} \right\}_{k=1}^{p},\quad {\mathcal{W}}^{\prime} = \left\{ w_{l}^{\prime} \right\}_{l=1}^{q} $$

그러면 두 선형변환 $\phi$, $\psi$의 행렬표현이 다음과 같이 존재한다.

$$ A = \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}}\in M_{m \times n} \qquad B = \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}} \in M_{q \times p} $$

텐서곱 $V \otimes W$의 순서기저를 $\mathcal{V} \otimes \mathcal{W} = \left\{ v_{i} \otimes w_{k} \right\}$과 같이 표기하고, 순서를 다음과 같이 주자.

$$ v_{1} \otimes w_{1}, \dots, v_{1} \times w_{p}, \\ v_{2}\otimes w_{1}, \dots, v_{2} \times w_{p}, \\ \dots \\ v_{n}\otimes w_{1}, \dots, v_{n} \times w_{p} $$

$V^{\prime} \otimes W^{\prime}$의 기저 $\mathcal{V}^{\prime} \otimes \mathcal{W}^{\prime} = \left\{ v_{j}^{\prime} \otimes w_{l}^{\prime} \right\}$에도 같은 식으로 순서를 주자. 그러면 $\phi$와 $\psi$의 텐서곱 또한 $\phi \otimes \psi : V \otimes W \to V^{\prime} \otimes W^{\prime}$인 선형변환이므로 다음과 같은 행렬표현matrix representation이 존재한다.

$$ \begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} $$

정리

두 선형변환 $\phi : V \to V^{\prime}$, $\psi : W \to W^{\prime}$의 행렬표현을 각각 $A = \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}}$, $B = \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}}$라 하자. 텐서곱 $\phi \otimes \psi : V \otimes W \to V^{\prime} \otimes W^{\prime}$의 행렬표현은 $A$와 $B$의 크로네커 곱과 같다.

$$ \begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} = A \otimes B = \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}} \otimes \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}} $$

증명

행렬표현 $\begin{bmatrix} \phi \otimes \psi \end{bmatrix}_{\mathcal{V} \otimes \mathcal{W}}^{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}}$를 찾기 위해서는 정의역의 기저 $\mathcal{V} \otimes \mathcal{W}$가 $\phi \otimes \psi$에 의해 어떻게 매핑되는지를 보면 된다. 우선 두 선형변환의 행렬표현을 다음과 같다고 하자.

$$ \begin{bmatrix} \phi \end{bmatrix}_{\mathcal{V}}^{{\mathcal{V}}^{\prime}} = A = [ \alpha_{ji} ] \in M_{m \times n} \qquad \begin{bmatrix} \psi \end{bmatrix}_{\mathcal{W}}^{{\mathcal{W}}^{\prime}} = B = [ \beta_{lk} ] \in M_{q \times p} $$

다시말해 $\phi(v_{i}) = \sum\limits_{j}\alpha_{ji}v_{j}^{\prime}$, $\psi(w_{k}) = \sum\limits_{l}\beta_{lk}w_{k}^{\prime}$이다. 선형변환의 텐서곱곱 벡터의 정의에 의해 기저벡터 $v_{i} \otimes w_{k}$는 다음과 같이 매핑된다.

$$ \begin{align*} (\phi \otimes \psi)(v_{i} \otimes w_{k}) &= \phi(v_{i}) \otimes \psi(w_{k}) \\ &= \left( \sum\limits_{j}\alpha_{ji}v_{j}^{\prime} \right) \otimes \left( \sum\limits_{l}\beta_{lk}w_{l}^{\prime} \right) \\ &= \sum_{j,l} \alpha_{ji}\beta_{lk} v_{j}^{\prime} \otimes w_{l}^{\prime} \\ \end{align*} $$

$$ \implies \left[ (\phi \otimes \psi)(v_{i} \otimes w_{k}) \right]_{{\mathcal{V}}^{\prime} \otimes \mathcal{W}^{\prime}} = \begin{bmatrix} \alpha_{1i}\beta_{1k} \\ \alpha_{1i}\beta_{2k} \\ \vdots \\ \alpha_{1i}\beta_{qk} \\ \alpha_{2i}\beta_{1k} \\ \alpha_{2i}\beta_{2k} \\ \vdots \\ \alpha_{2i}\beta_{qk} \\ \vdots \\ \alpha_{mi}\beta_{1k} \\ \alpha_{mi}\beta_{2k} \\ \vdots \\ \alpha_{mi}\beta_{qk} \\ \end{bmatrix} $$

따라서 정리하면 다음과 같다.

$$ \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*} $$


  1. 김영훈·허재성, 양자 정보 이론 (2020), p36 ↩︎

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