I think this might be an odd question, and a little vague. But here goes.
This is related to coordinate transformations. Three matrices are given: $G_1 , G_2$, and $\Lambda$. $G_1$ and $G_2$ are symmetric. (They are metrics, actually.) $\Lambda$ is the matrix in question. The three matrices satisfy
$\Lambda^{-1} = G_1^{-1} \Lambda^T G_2$
Is there anything we can say about the properties of $\Lambda$, such as it is symmetric, or it is either symteric or anti-symmetric, or ...?
Thanks.
If you rewrite this as $$G_1 = \Lambda^T G_2 \Lambda\,,$$ this is just the change-of-basis formula for a symmetric bilinear form. That is, $\Lambda$ is the matrix that interpolates from the expression for the metric in one coordinate system to the expression in another coordinate system. As it stands, $\Lambda$ could be any invertible matrix.