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：
\[
\begin{align*}
{\bf u'}_i \otimes {\bf v'}_j
&=
\left( \sum_{k} a^k_i {\bf u}_k \right)
\otimes
\left( \sum_{k} c^k_j {\bf v}_k \right)
\\
&=
\sum_{k,l}
\left( a^k_i {\bf u}_k \right)
\otimes
\left( c^l_j {\bf v}_l \right)
\\
&=
\sum_{k,l}
\left( a^k_i c^l_j \right)
\left( {\bf u}_k \otimes {\bf v}_l \right)
\\
&=
\sum_{k}
a^k_i
\left(
\sum_{l}
c^l_j ( {\bf u}_k \otimes {\bf v}_l )
\right)
\end{align*}
\]
\[
\left(
\begin{array}{ccc}
{\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\
& \cdots & \\
{\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\
\end{array}
\right)
\\
=
\left(
\begin{array}{ccc}
a^1_1 & \cdots & a^n_1 \\
& \cdots & \\
a^1_n & \cdots & a^n_n \\
\end{array}
\right)
\left(
\begin{array}{ccc}
{\sum_{l} c^l_1 {\bf u}_1 \otimes {\bf v}_l }
& \cdots &
{\sum_{l} c^l_m {\bf u}_1 \otimes {\bf v}_l }
\\
& \cdots &
\\
{\sum_{l} c^l_1 {\bf u}_n \otimes {\bf v}_l }
& \cdots &
{\sum_{l} c^l_m {\bf u}_n \otimes {\bf v}_l }
\\
\end{array}
\right)
\\
=
\left(
\begin{array}{ccc}
a^1_1 & \cdots & a^n_1 \\
& \cdots & \\
a^1_n & \cdots & a^n_n \\
\end{array}
\right)
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
\left(
\begin{array}{ccc}
c_1^1 & \cdots & c_m^1 \\
& \cdots & \\
c_1^m & \cdots & c_m^m \\
\end{array}
\right)
\\
=
{}^t A\
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
\ C
\]
\[
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
\\=
{}^t (AB)
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
(CD)
\\=
{}^t B
\left(
{}^t A
\left(
\begin{array}{ccc}
{\bf u}_1 \otimes {\bf v}_1 & \cdots & {\bf u}_1 \otimes {\bf v}_m \\
& \cdots & \\
{\bf u}_n \otimes {\bf v}_1 & \cdots & {\bf u}_n \otimes {\bf v}_m \\
\end{array}
\right)
C
\right)
D
\\=
{}^t B
\left(
\begin{array}{ccc}
{\bf u'}_1 \otimes {\bf v'}_1 & \cdots & {\bf u'}_1 \otimes {\bf v'}_m \\
& \cdots & \\
{\bf u'}_n \otimes {\bf v'}_1 & \cdots & {\bf u'}_n \otimes {\bf v'}_m \\
\end{array}
\right)
D
\]
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