I am trying, line by line, to find the general form of a $2\times 2$ matrix that leaves the square of a complex number invariant.
$$ z=a+ib \implies z^2 = a^2-b^2+2iab $$
To do that, I consider $z$ to be a vector $(a,b)$, and $T$ to be a $2\times 2$ matrix, as follows:
$$ (Tz)^2=a^2-b^2+2iab $$
One can do it the easy way by noting the z commutes with $T$ and therefore $T^2z^2=a^2-b^2+2iab\implies T^2=I$. The product-preserving set of linear transformations is the set of all complex matrices squaring to the identity matrix.
But I am trying to do it without shortcuts to learn better. My starting point is:
$$ \pmatrix{T_{00} & T_{01} \\ T_{10} & T_{11}} \pmatrix{a\\b}=T_{00}a+T_{01}b+T_{10}ia+T_{11}ib $$
Squaring $Tz$, I get:
$$ (Tz)^2=a^2 T_{00}^2+2 i a^2 T_{00} T_{10}-a^2 T_{10}^2+2 i a b (T_{00} T_{11}+T_{01} T_{10})+2 a b T_{00} T_{01}-2 a b T_{10} T_{11}+b^2 T_{01}^2+2 i b^2 T_{01} T_{11}-b^2 T_{11}^2 $$
Now, to solve for $T_{00},T_{01},T_{10},T_{11}$, I must find 4 independent equations from the relations above, correct? What is the trick to identify those? If the following logic sound:
I equate $z^2$ to $(Tz)^2$:
$$ a^2-b^2+2iab=a^2 T_{00}^2+2 i a^2 T_{00} T_{10}-a^2 T_{10}^2+2 i a b T_{00} T_{11}+2 i a b T_{01} T_{10}+2 a b T_{00} T_{01}-2 a b T_{10} T_{11}+b^2 T_{01}^2+2 i b^2 T_{01} T_{11}-b^2 T_{11}^2 $$
I need to the real part (on the left) to equal the real part (on the right), and the imaginary part (on the left) to equal the imaginary part (on the right). That gives two equations:
First the real part:
$$ \tag{real} a^2-b^2=a^2 T_{00}^2-a^2 T_{10}^2+2 a b T_{00} T_{01}-2 a b T_{10} T_{11}+b^2 T_{01}^2-b^2 T_{11}^2 $$
gives
$$ \begin{align} a^2 &= a^2T_{00}^2- a^2T_{10}^2 && (1) \\ -b^2 &=b^2T_{01}^2- b^2T_{11}^2 && (2)\\ 0 &=+2 a b T_{00} T_{01}-2 a b T_{10} T_{11} && (3) \end{align} $$
now the imaginary part:
$$ \tag{imaginary} 2iab=2 i a^2 T_{00} T_{10}+2 i a b T_{00} T_{11}+2 i a b T_{01} T_{10}+2 i b^2 T_{01} T_{11} $$
gives:
$$ \begin{align} 2iab &=2 i a b T_{00} T_{11}+2 i a b T_{01} T_{10} && (4)\\ 0 &=2 i a^2 T_{00} T_{10} && (5)\\ 0 &=2 i b^2 T_{01} T_{11} && (6) \end{align} $$
$$ \begin{align} a^2 &= a^2T_{00}^2- a^2T_{10}^2 && (1) \\ -b^2 &=b^2T_{01}^2- b^2T_{11}^2 && (2)\\ 0 &=+2 a b T_{00} T_{01}-2 a b T_{10} T_{11} && (3) \end{align} $$
Are these 6 equations valid constraints?
I do not know from here how to get to $T^1=I$?