This is a long multi-steps question and I'm stuck at the last leg. I believe my question to be trivial but after 3 hrs of staring and trying all sort of methods (ridiculous ones even) I'm not getting anywhere. Frustrating!
I shall jump straight to the problem I am facing and provide prior information leading to this.
I do not understand how I can arrive at the corresponding matrix field in the last line.
The prior worked out basis products are:
The given polar coordinates are
In Euclidean coordinate, Q and R
I really need help and appreciate any.
Thanks in advance
You already have the basis for the matrices; it is $\{\boldsymbol{e_{1}}\otimes\boldsymbol{e^{1}}, \boldsymbol{e_{1}}\otimes\boldsymbol{e^{2}},\boldsymbol{e_{2}}\otimes\boldsymbol{e^{1}}, \boldsymbol{e_{2}}\otimes\boldsymbol{e^{2}}\}$. This means that a matrix $\boldsymbol{M}$, under this basis, given by $$\boldsymbol{M}=\left(\begin{array}{cc} a & b\\ c & d \end{array}\right),$$ is $$\boldsymbol{M}= a\,(\boldsymbol{e_{1}}\otimes\boldsymbol{e^{1}}) + b\,(\boldsymbol{e_{1}}\otimes\boldsymbol{e^{2}}) + c\,(\boldsymbol{e_{2}}\otimes\boldsymbol{e^{1}})+d\,(\boldsymbol{e_{2}}\otimes\boldsymbol{e^{2}}).$$