I have a mapping $f:R^3\rightarrow R^3$ such that: $$x_1=(1+\eta)X_1$$ $$x_2=(1+\gamma)X_2$$ $$x_3=(1+\gamma)X_3$$ and I have to obtain the transformation matrix $\varepsilon$ corresponding to this mapping. I know the matrix has to be$$\varepsilon=\begin{pmatrix}1+\eta& 0& 0\\ 0& 1+\gamma& 0\\ 0& 0& 1+\gamma\end{pmatrix}.$$ I want to get this matrix using Mathematica but I don't know how. I tried the following:
x1 = (1 + \[Eta]) X1;
x2 = (1 + \[Gamma]) X2;
x3 = (1 + \[Gamma]) X3;
\[Epsilon] = {{\[Epsilon]11, \[Epsilon]12, \[Epsilon]13}, \
{\[Epsilon]21, \[Epsilon]22, \[Epsilon]23}, {\[Epsilon]31, \
\[Epsilon]32, \[Epsilon]33}};
Solve[{x1, x2, x3} == \[Epsilon] .{X1, X2, X3}, {\[Epsilon]11, \[Epsilon]12,
\[Epsilon]13, \[Epsilon]21, \[Epsilon]22, \[Epsilon]23, \[Epsilon]31,
\[Epsilon]32, \[Epsilon]33}] // Simplify
but I got the message Solve::svars: Equations may not give solutions for all "solve" variables.
and some strange output.
How to let Mathematica correctly solve for $\varepsilon$ ?
sol
be your solution and try(\[Epsilon] /. sol).{X1, X2, X3} // Simplify
. Basically, it means that you're free to set 6 of your variables to whatever you want (like zero). $\endgroup$