Catatan Tambahan Tensor dan Matriks

Tinjau transformasi kordinat berupa rotasi sumbu kordinat. Dengan transformasi ini suatu vektor $\vec{U}=U_{1}\hat{e_{1}}+U_{2}\hat{e_{2}}+U_{3}\hat{e_{3}}$ menjadi vektor $\vec{U}'=U_{1}'\hat{e_{1}'}+U_{2}'\hat{e_{2}'}+U_{3}'\hat{e_{3}'}$ dan vektor $\vec{V}=V_{1}\hat{e_{1}}+V_{2}\hat{e_{2}}+V_{3}\hat{e_{3}}$ menjadi vektor $\vec{V}'=V_{1}'\hat{e_{1}'}+V_{2}'\hat{e_{2}'}+V_{3}'\hat{e_{3}'}$.
Dalam notasi matriks, hal ini dapat dinyatakan dalam bentuk
\begin{eqnarray}
\left(\begin{array}{c}
U_{1}'\\
U_{2}'\\
U_{3}'
\end{array}\right) & = & A\left(\begin{array}{c}
U_{1}\\
U_{2}\\
U_{3}
\end{array}\right)\nonumber \\
 & = & \left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{array}\right)\left(\begin{array}{c}
U_{1}\\
U_{2}\\
U_{3}
\end{array}\right)\nonumber \\
 & = & \left(\begin{array}{c}
a_{11}U_{1}+a_{12}U_{2}+a_{13}U_{3}\\
a_{21}U_{1}+a_{22}U_{2}+a_{23}U_{3}\\
a_{31}U_{1}+a_{32}U_{2}+a_{33}U_{3}
\end{array}\right)\label{eq:matriks-u-aksen}
\end{eqnarray}
\begin{eqnarray}
\left(\begin{array}{c}
V_{1}'\\
V_{2}'\\
V_{3}'
\end{array}\right) & = & A\left(\begin{array}{c}
V_{1}\\
V_{2}\\
V_{3}
\end{array}\right)=\left(\begin{array}{c}
a_{11}V_{1}+a_{12}V_{2}+a_{13}V_{3}\\
a_{21}V_{1}+a_{22}V_{2}+a_{23}V_{3}\\
a_{31}V_{1}+a_{32}V_{2}+a_{33}V_{3}
\end{array}\right)\label{eq:matriks-v-aksen}
\end{eqnarray}
Telah dipahami pula bahwa direct product dari dua buah vektor menghasilkan suatu tensor rank 2
\[
\vec{r}\otimes\vec{p}=\left(\begin{array}{c}
r_{1}\\
r_{2}\\
r_{3}
\end{array}\right)\otimes\left(\begin{array}{ccc}
p_{1} & p_{2} & p_{3}\end{array}\right)=\left(\begin{array}{ccc}
r_{1}p_{1} & r_{1}p_{2} & r_{1}p_{3}\\
r_{2}p_{1} & r_{2}p_{2} & r_{2}p_{3}\\
r_{3}p_{1} & r_{3}p_{2} & r_{3}p_{3}
\end{array}\right)
\]
Jika tensor tersebut dinyatakan sebagai tensor $B$, maka
\begin{eqnarray*}
B & = & \left(\begin{array}{ccc}
b_{11} & b_{12} & b_{13}\\
b_{21} & b_{22} & b_{23}\\
b_{31} & b_{32} & b_{33}
\end{array}\right)=\left(\begin{array}{ccc}
r_{1}p_{1} & r_{1}p_{2} & r_{1}p_{3}\\
r_{2}p_{1} & r_{2}p_{2} & r_{2}p_{3}\\
r_{3}p_{1} & r_{3}p_{2} & r_{3}p_{3}
\end{array}\right)
\end{eqnarray*}
Artinya jika dinyatakan dalam bentuk komponen, maka komponen baris ke-$i$ kolom ke-$j$ dari matriks $B$ adalah
\[
b_{ij}=r_{i}p_{j}
\]
Jadi direct product dari $\vec{U}'$ dan $\vec{V}'$ juga menghasilkan suatu tensor rank 2
\begin{eqnarray*}
\vec{U}'\otimes\vec{V}' & = & \left(\begin{array}{c}
U_{1}'\\
U_{2}'\\
U_{3}'
\end{array}\right)\otimes\left(\begin{array}{ccc}
V_{1}' & V_{2}' & V_{3}'\end{array}\right)\\
 & = & \left(\begin{array}{ccc}
U_{1}'V_{1}' & U_{1}'V_{2}' & U_{1}'V_{3}'\\
U_{2}'V_{1}' & U_{2}'V_{2}' & U_{2}'V_{3}'\\
U_{3}'V_{1}' & U_{3}'V_{2}' & U_{3}'V_{3}'
\end{array}\right)
\end{eqnarray*}
Kemudian dengan menggunakan hasil pada persamaan \ref{eq:matriks-u-aksen} dan persamaan \ref{eq:matriks-v-aksen}, maka diperoleh
\begin{eqnarray*}
\vec{U}'\otimes\vec{V}' & = & \left(\begin{array}{ccc}
U_{1}'V_{1}' & U_{1}'V_{2}' & U_{1}'V_{3}'\\
U_{2}'V_{1}' & U_{2}'V_{2}' & U_{2}'V_{3}'\\
U_{3}'V_{1}' & U_{3}'V_{2}' & U_{3}'V_{3}'
\end{array}\right)\equiv W'
\end{eqnarray*}
\begin{eqnarray*}
W_{11}' & = & U_{1}'V_{1}'=(a_{11}U_{1}+a_{12}U_{2}+a_{13}U_{3})(a_{11}V_{1}+a_{12}V_{2}+a_{13}V_{3})\\
 & = & \quad a_{11}a_{11}U_{1}V_{1}+a_{11}a_{12}U_{1}V_{2}+a_{11}a_{13}U_{1}V_{3}\\
 &  & +\,a_{12}a_{11}U_{2}V_{1}+a_{12}a_{12}U_{2}V_{2}+a_{12}a_{13}U_{2}V_{3}\\
 &  & +\,a_{13}a_{11}U_{3}V_{1}+a_{13}a_{12}U_{3}V_{2}+a_{13}a_{13}U_{3}V_{3}\\
 & = & \underset{j=1}{\overset{3}{\sum}}a_{11}a_{1j}U_{1}V_{j}+\underset{j=1}{\overset{3}{\sum}}a_{12}a_{1j}U_{2}V_{j}+\underset{j=1}{\overset{3}{\sum}}a_{13}a_{1j}U_{1}V_{j}\\
 & = & \underset{i=1}{\overset{3}{\sum}}\left(\underset{j=1}{\overset{3}{\sum}}a_{1i}a_{1j}U_{i}V_{j}\right)=\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{1i}a_{1j}U_{i}V_{j}
\end{eqnarray*}
\begin{eqnarray*}
W_{12}' & = & U_{1}'V_{2}'=(a_{11}U_{1}+a_{12}U_{2}+a_{13}U_{3})(a_{21}V_{1}+a_{22}V_{2}+a_{23}V_{3})\\
 & = & \quad a_{11}a_{21}U_{1}V_{1}+a_{11}a_{22}U_{1}V_{2}+a_{11}a_{23}U_{1}V_{3}\\
 &  & +\,a_{12}a_{21}U_{2}V_{1}+a_{12}a_{22}U_{2}V_{2}+a_{12}a_{23}U_{2}V_{3}\\
 &  & +\,a_{13}a_{21}U_{3}V_{1}+a_{13}a_{22}U_{3}V_{2}+a_{13}a_{23}U_{3}V_{3}\\
 & = & \underset{j=1}{\overset{3}{\sum}}a_{11}a_{2j}U_{1}V_{j}+\underset{j=1}{\overset{3}{\sum}}a_{12}a_{2j}U_{2}V_{j}+\underset{j=1}{\overset{3}{\sum}}a_{13}a_{2j}U_{3}V_{j}\\
 & = & \underset{i=1}{\overset{3}{\sum}}\left(\underset{j=1}{\overset{3}{\sum}}a_{1i}a_{2j}U_{i}V_{j}\right)=\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{1i}a_{2j}U_{i}V_{j}
\end{eqnarray*}
Dapat diteruskan untuk memperoleh komponen tensor $W'$ lainnya
\begin{align*}
W_{13}' & =\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{1i}a_{3j}U_{i}V_{j}\\
W_{21}' & =\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{2i}a_{1j}U_{i}V_{j}\\
 & \ldots
\end{align*}
Bila digeneralisasi, maka akan diperoleh
\[
W_{\alpha\beta}'=\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{\alpha i}a_{\beta j}U_{i}V_{j}
\]
dan jika $W_{ij}=U_{i}V_{j}$, maka
\[
W_{\alpha\beta}'=\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{\alpha i}a_{\beta j}W_{ij}
\]
Karena $W_{\alpha\beta}'=U_{\alpha}'V_{\beta}'$, maka dapat pula dinyatakan
\[
W_{\alpha\beta}'=U_{\alpha}'V_{\beta}'=\underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}a_{\alpha i}a_{\beta j}U_{i}V_{j}
\]
Hal yang sama dapat digeneralisasi untuk memperoleh tensor rank yang lebih tinggi, misalnya untuk tensor rank 4:
\[
\begin{array}{ccl}
W_{\alpha\beta\gamma\delta}'=M_{\alpha}'N_{\beta}'O_{\gamma}'P_{\delta}' & = & \underset{i=1}{\overset{3}{\sum}}\underset{j=1}{\overset{3}{\sum}}\underset{k=1}{\overset{3}{\sum}}\underset{l=1}{\overset{3}{\sum}}a_{\alpha i}a_{\beta j}a_{\gamma k}a_{\delta l}W_{ijkl}\\
 & = & \underset{i,j,k,l=1}{\overset{3}{\sum}}a_{\alpha i}a_{\beta j}a_{\gamma k}a_{\delta l}W_{ijkl}
\end{array}
\]
Jika menggunakan notasi konvensi penjumlahan (\emph{summation convention})
\[
W_{\alpha\beta\gamma\delta}'=a_{\alpha i}a_{\beta j}a_{\gamma k}a_{\delta l}W_{ijkl}
\]