I have this problem: Let $A = \begin{pmatrix} 2i & 1 \\ 1 & 0 \end{pmatrix}.$ Find a $N\in\mathbb GL(3,\mathbb C) $ such that $N^{t}A\overline N$ is diagonal
The matrix A is $2\ x\ 2$ so I have tried with a certain $N = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ . My professor tried with this matrix and said $N^{t}A\overline N$= $NA\overline N{^t}$ why?
After he has obtained the following system of equation: $2ia\overline a+b\overline a+a\overline b=0$ and $2ic\overline c+d\overline c+c\overline d=0$ but how do I find this coefficient $a,b,c,d$ ?
Thanks for help.