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:
\[
( {\bf u'}_1, \cdots, {\bf u'}_m )
= ( {\bf u}_1, \cdots, {\bf u}_m ) \ A \\
( {\bf u}_1, \cdots, {\bf u}_m )
= ( {\bf u'}_1, \cdots, {\bf u'}_m ) \ B
\\ \\
\quad
A =
\left(
\begin{array}{ccc}
a_1^1 & \cdots & a_m^1 \\
& \cdots & \\
a_1^m & \cdots & a_m^m \\
\end{array}
\right)
\qquad
B = A^{-1} =
\left(
\begin{array}{ccc}
b_1^1 & \cdots & b_m^1 \\
& \cdots & \\
b_1^m & \cdots & b_m^m \\
\end{array}
\right)
\\ \\
( {\bf v'}_1, \cdots, {\bf v'}_n )
= ( {\bf v}_1, \cdots, {\bf v}_n ) \ C \\
( {\bf v}_1, \cdots, {\bf v}_n )
= ( {\bf v'}_1, \cdots, {\bf v'}_n) \ D
\\ \\
\quad
C =
\left(
\begin{array}{ccc}
c_1^1 & \cdots & c_n^1 \\
& \cdots & \\
c_1^n & \cdots & c_n^n \\
\end{array}
\right)
\qquad
D = C^{-1} =
\left(
\begin{array}{ccc}
d_1^1 & \cdots & d_n^1 \\
& \cdots & \\
d_1^n & \cdots & d_n^n \\
\end{array}
\right)
\]
\[
\left(
\begin{array}{ccc}
{\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\
& \cdots & \\
{\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\
\end{array}
\right)
=
{}^t A\
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
\ C
\\ \\
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
=
{}^t B
\left(
\begin{array}{ccc}
{\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\
& \cdots & \\
{\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\
\end{array}
\right)
D
\]
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,
\[
{}^t ( {x'}^1, \cdots, {x'}^m ) = B\ {}^t( {x}^1, \cdots, {x}^m )
\\ {}^t( {y'}^1, \cdots, {y'}^n ) = D\ {}^t( {y}^1, \cdots, {y}^n )
\\ \\
\begin{align*}
{x'}^i {y'}^j
&=
\left( \sum_{k} b^i_k {x}^k \right)
\left( \sum_{k} d^j_k {y}^k \right)
\\
&=
\sum_{k,l}
\left( b^i_k {x}^k \right)
\left( d^j_l {y}^l \right)
\\
&=
\sum_{k,l}
\left( b^i_k d^j_l \right)
\left( {x}^k {y}^l \right)
\\
&=
\sum_{k}
b^i_k
\left(
\sum_{l}
d^j_l ( {x}^k {y}^l )
\right)
\end{align*}
\]
\[
\left(
\begin{array}{ccc}
{x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^n \\
& \cdots & \\
{x'}^m {y'}^1 & \cdots & {x'}^m {y'}^n \\
\end{array}
\right)
\\
=
\left(
\begin{array}{ccc}
b^1_1 & \cdots & b^1_m \\
& \cdots & \\
b^m_1 & \cdots & b^m_m \\
\end{array}
\right)
\left(
\begin{array}{ccc}
{\sum_{l} d^1_l {x}^1 {y}^l }
& \cdots &
{\sum_{l} d^n_l {x}^1 {y}^l }
\\
& \cdots &
\\
{\sum_{l} d^1_l {x}^m {y}^l }
& \cdots &
{\sum_{l} d^n_l {x}^m {y}^l }
\\
\end{array}
\right)
\\
=
\left(
\begin{array}{ccc}
b^1_1 & \cdots & b^1_m \\
& \cdots & \\
b^m_1 & \cdots & b^m_m \\
\end{array}
\right)
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
\left(
\begin{array}{ccc}
d^1_1 & \cdots & d^n_1 \\
& \cdots & \\
d^1_n & \cdots & d^n_n \\
\end{array}
\right)
\\
=
B\
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
\ {}^t D
\]
\[
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
\\=
A B
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
{}^t (C D)
\\=
A
\left(
B
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
{}^t D
\right)
{}^t C
\\=
A\
\left(
\begin{array}{ccc}
{x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^m \\
& \cdots & \\
{x'}^n {y'}^1 & \cdots & {x'}^n {y'}^m \\
\end{array}
\right)
\ {}^t C
\]
まとめ:
\[
\left(
\begin{array}{ccc}
{x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^n \\
& \cdots & \\
{x'}^m {y'}^1 & \cdots & {x'}^m {y'}^n \\
\end{array}
\right)
=
B\
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
\ {}^t D
\\ \\
\left(
\begin{array}{ccc}
{x}^1 {y}^1 & \cdots & {x}^1 {y}^n \\
& \cdots & \\
{x}^m {y}^1 & \cdots & {x}^m {y}^n \\
\end{array}
\right)
=
A\
\left(
\begin{array}{ccc}
{x'}^1 {y'}^1 & \cdots & {x'}^1 {y'}^m \\
& \cdots & \\
{x'}^n {y'}^1 & \cdots & {x'}^n {y'}^m \\
\end{array}
\right)
\ {}^t C
\]
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