Up 座標変換の式 作成: 2018-01-23
更新: 2018-02-24


    線型空間
      \( U\) : 体 \(K\) 上 \(m\) 次元
      \( V\) : 体 \(K\) 上 \(n\) 次元
    のテンソル積 \( U \otimes V\) の基底変換 \[ \{ {\bf u}_i \otimes {\bf v}_j \} \to \{ {\bf u'}_i \otimes {\bf v'}_j \} \\ \begin{align} \qquad & {\bf u} = \{ {\bf u}_1, \cdots, {\bf u}_m \}, & {\bf v} = \{ {\bf v}_1, \cdots, {\bf v}_n \} \\ \qquad & {\bf u’} = \{ {\bf u'}_1, \cdots, {\bf u'}_m \}, & {\bf v'} = \{ {\bf v'}_1, \cdots, {\bf v'}_n \} \end{align} \] が,つぎのように得られた:
    \[ ( {\bf u'}_1, \cdots, {\bf u'}_m ) = ( {\bf u}_1, \cdots, {\bf u}_m ) \ A \\ ( {\bf u}_1, \cdots, {\bf u}_m ) = ( {\bf u'}_1, \cdots, {\bf u'}_m ) \ B \\  \\ \quad A = \left( \begin{array}{ccc} a_1^1 & \cdots & a_m^1 \\ & \cdots & \\ a_1^m & \cdots & a_m^m \\ \end{array} \right) \qquad B = A^{-1} = \left( \begin{array}{ccc} b_1^1 & \cdots & b_m^1 \\ & \cdots & \\ b_1^m & \cdots & b_m^m \\ \end{array} \right) \\  \\ ( {\bf v'}_1, \cdots, {\bf v'}_n ) = ( {\bf v}_1, \cdots, {\bf v}_n ) \ C \\ ( {\bf v}_1, \cdots, {\bf v}_n ) = ( {\bf v'}_1, \cdots, {\bf v'}_n) \ D \\  \\ \quad C = \left( \begin{array}{ccc} c_1^1 & \cdots & c_n^1 \\ & \cdots & \\ c_1^n & \cdots & c_n^n \\ \end{array} \right) \qquad D = C^{-1} = \left( \begin{array}{ccc} d_1^1 & \cdots & d_n^1 \\ & \cdots & \\ d_1^n & \cdots & d_n^n \\ \end{array} \right) \]
      のとき,
    \[ \left( \begin{array}{ccc} {\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\ & \cdots & \\ {\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\ \end{array} \right) = {}^t A\ \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) \ C \\   \\ \left( \begin{array}{ccc} {\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\ & \cdots & \\ {\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\ \end{array} \right) = {}^t B \left( \begin{array}{ccc} {\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\ & \cdots & \\ {\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\ \end{array} \right) D \]

    続いて,この基底変換に応じる座標変換の式を求める。
    つぎのように設定する:
      \({\bf x} \in U \)
        基底 \( \{{\bf u}_i \} \) に対する座標が \( ( {x}^1, \cdots, {x}^m )\)
        基底 \( \{{\bf u'}_i \} \) に対する座標が \( ( {x'}^1, \cdots, {x'}^m )\)
      \( {\bf y} \in V \)
        基底 \( \{{\bf v}_i \} \) に対する座標が \( ( {y}^1, \cdots, {y}^n )\)
        基底 \( \{{\bf v'}_i \} \) に対する座標が \( ( {y'}^1, \cdots, {y'}^n )\)
    このとき, \[ {\bf x} \otimes {\bf y} = \sum_{i,j} (x^i {y}^j ) ( {\bf u}_i \otimes {\bf v}_j ) = \sum_{i,j} ({x'}^i {y'}^j ) ( {\bf u'}_i \otimes {\bf v'}_j ) \] そして, \[ {}^t ( {x'}^1, \cdots, {x'}^m ) = B\ {}^t( {x}^1, \cdots, {x}^m ) \\ {}^t( {y'}^1, \cdots, {y'}^n ) = D\ {}^t( {y}^1, \cdots, {y}^n ) \\   \\ \begin{align*} {x'}^i {y'}^j &= \left( \sum_{k} b^i_k {x}^k \right) \left( \sum_{k} d^j_k {y}^k \right) \\ &= \sum_{k,l} \left( b^i_k {x}^k \right) \left( d^j_l {y}^l \right) \\ &= \sum_{k,l} \left( b^i_k d^j_l \right) \left( {x}^k {y}^l \right) \\ &= \sum_{k} b^i_k \left( \sum_{l} d^j_l ( {x}^k {y}^l ) \right) \end{align*} \]
      よって,
    \[ \left( \begin{array}{ccc} {x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^n \\ & \cdots & \\ {x'}^m {y'}^1 & \cdots & {x'}^m {y'}^n \\ \end{array} \right) \\ = \left( \begin{array}{ccc} b^1_1 & \cdots & b^1_m \\ & \cdots & \\ b^m_1 & \cdots & b^m_m \\ \end{array} \right) \left( \begin{array}{ccc} {\sum_{l} d^1_l {x}^1 {y}^l } & \cdots & {\sum_{l} d^n_l {x}^1 {y}^l } \\ & \cdots & \\ {\sum_{l} d^1_l {x}^m {y}^l } & \cdots & {\sum_{l} d^n_l {x}^m {y}^l } \\ \end{array} \right) \\ = \left( \begin{array}{ccc} b^1_1 & \cdots & b^1_m \\ & \cdots & \\ b^m_1 & \cdots & b^m_m \\ \end{array} \right) \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) \left( \begin{array}{ccc} d^1_1 & \cdots & d^n_1 \\ & \cdots & \\ d^1_n & \cdots & d^n_n \\ \end{array} \right) \\ = B\ \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) \ {}^t D \]
      また,これより
    \[ \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) \\= A B \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) {}^t (C D) \\= A \left( B \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) {}^t D \right) {}^t C \\= A\ \left( \begin{array}{ccc} {x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^m \\ & \cdots & \\ {x'}^n {y'}^1 & \cdots & {x'}^n {y'}^m \\ \end{array} \right) \ {}^t C \]
    まとめ: \[ \left( \begin{array}{ccc} {x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^n \\ & \cdots & \\ {x'}^m {y'}^1 & \cdots & {x'}^m {y'}^n \\ \end{array} \right) = B\ \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) \ {}^t D \\   \\ \left( \begin{array}{ccc} {x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\ & \cdots & \\ {x}^m {y}^1 & \cdots & {x}^m {y}^n \\ \end{array} \right) = A\ \left( \begin{array}{ccc} {x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^m \\ & \cdots & \\ {x'}^n {y'}^1 & \cdots & {x'}^n {y'}^m \\ \end{array} \right) \ {}^t C \]