Up 基底変換の式 作成: 2018-01-23
更新: 2018-02-24


    テンソル積の座標は,基底に対して決まる。
    言い換えると,基底変換には座標変換が応じる。
    以下,テンソル積の基底変換について。


    簡単のため,2つの線型空間のテンソル積で考える。

    線型空間
      \( U\) : 体 \(K\) 上 \(m\) 次元
      \( V\) : 体 \(K\) 上 \(n\) 次元
    のテンソル積 \( U \otimes V\) の基底は,\( U,\, V\) それぞれの基底 \[ {\bf u} = \{ {\bf u}_1, \cdots, {\bf u}_m \} \\ {\bf v} = \{ {\bf v}_1, \cdots, {\bf v}_n \} \] に対する \[ \{\ {\bf u}_i \otimes {\bf v}_j \ |\ i = 1,\cdots,m;\ j = 1,\cdots,n \ \} \] である。

    そこで \( U \otimes V\) における基底変換は,\( U,\, V\) の基底 \[ {\bf u’} = \{ {\bf u'}_1, \cdots, {\bf u'}_m \} \\ {\bf v'} = \{ {\bf v'}_1, \cdots, {\bf v'}_n \} \] が加わったときの,基底 \(\{ {\bf u}_i \otimes {\bf v}_j \}\) から基底 \[ \{ {\bf u'}_i \otimes {\bf v'}_j \} \] への変換である。


    つぎのように措く:
    \[ ( {\bf u'}_1, \cdots, {\bf u'}_m ) = ( {\bf u}_1, \cdots, {\bf u}_m ) \ A \\ ( {\bf u}_1, \cdots, {\bf u}_m ) = ( {\bf u'}_1, \cdots, {\bf u'}_m ) \ B \\  \\ \quad A = \left( \begin{array}{ccc} a_1^1 & \cdots & a_m^1 \\ & \cdots & \\ a_1^m & \cdots & a_m^m \\ \end{array} \right) \qquad B = \left( \begin{array}{ccc} b_1^1 & \cdots & b_m^1 \\ & \cdots & \\ b_1^m & \cdots & b_m^m \\ \end{array} \right) \\  \\ ( {\bf v'}_1, \cdots, {\bf v'}_n ) = ( {\bf v}_1, \cdots, {\bf v}_n ) \ C \\ ( {\bf v}_1, \cdots, {\bf v}_n ) = ( {\bf v'}_1, \cdots, {\bf v'}_n) \ D \\  \\ \quad C = \left( \begin{array}{ccc} c_1^1 & \cdots & c_n^1 \\ & \cdots & \\ c_1^n & \cdots & c_n^n \\ \end{array} \right) \qquad D = \left( \begin{array}{ccc} d_1^1 & \cdots & d_n^1 \\ & \cdots & \\ d_1^n & \cdots & d_n^n \\ \end{array} \right) \] このとき,
      \[ B = A^{-1} \\ D = C^{-1} \]


    基底の変換式を求める: \[ \begin{align*} {\bf u'}_i \otimes {\bf v'}_j &= \left( \sum_{k} a^k_i {\bf u}_k \right) \otimes \left( \sum_{k} c^k_j {\bf v}_k \right) \\ &= \sum_{k,l} \left( a^k_i {\bf u}_k \right) \otimes \left( c^l_j {\bf v}_l \right) \\ &= \sum_{k,l} \left( a^k_i c^l_j \right) \left( {\bf u}_k \otimes {\bf v}_l \right) \\ &= \sum_{k} a^k_i \left( \sum_{l} c^l_j ( {\bf u}_k \otimes {\bf v}_l ) \right) \end{align*} \]
      よって,
    \[ \left( \begin{array}{ccc} {\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\ & \cdots & \\ {\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\ \end{array} \right) \\ = \left( \begin{array}{ccc} a^1_1 & \cdots & a^n_1 \\ & \cdots & \\ a^1_n & \cdots & a^n_n \\ \end{array} \right) \left( \begin{array}{ccc} {\sum_{l} c^l_1 {\bf u}_1 \otimes {\bf v}_l } & \cdots & {\sum_{l} c^l_m {\bf u}_1 \otimes {\bf v}_l } \\ & \cdots & \\ {\sum_{l} c^l_1 {\bf u}_n \otimes {\bf v}_l } & \cdots & {\sum_{l} c^l_m {\bf u}_n \otimes {\bf v}_l } \\ \end{array} \right) \\ = \left( \begin{array}{ccc} a^1_1 & \cdots & a^n_1 \\ & \cdots & \\ a^1_n & \cdots & a^n_n \\ \end{array} \right) \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) \left( \begin{array}{ccc} c_1^1 & \cdots & c_m^1 \\ & \cdots & \\ c_1^m & \cdots & c_m^m \\ \end{array} \right) \\ = {}^t A\ \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) \ C \]
      そしてこれより,
    \[ \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) \\= {}^t (AB) \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) (CD) \\= {}^t B \left( {}^t A \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) C \right) D \\= {}^t B \left( \begin{array}{ccc} {\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\ & \cdots & \\ {\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\ \end{array} \right) D \]