Search for a constant Metric $\eta$ satisfying the a condition

Question

I am looking for a $2 \times 2$ constant metric $\eta$, satisfying the following condition:

$\eta A \eta^{-1}= A^T$.

where $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

Could anyone please help me.

Thanks in advance

Answer 1

Rearranging the equation $$\eqalign{ NAN^{-1} &= A^T \implies A^{-T}NA = N \cr }$$ and vectorizing both sides yields $$\eqalign{ {\rm vec}(N) &= (A^T\otimes A^{-T}){\,\rm vec}(N) \cr n &= Kn \cr }$$ So $n$ is any eigenvector of the Kronecker matrix $(K)$ corresponding to an eigenvalue of one,
and $N$ can be recovered by reshaping the eigenvector into a $2\times 2$ matrix.

The matrix $K$ and its eigenvalues/vectors will change with every $A$, so I don't see how the metric $(N)$ could possibly be constant for all $A$.