I've a doubt on othogonal matrices.
I know that an orthogonal matrix is a matrix $O$ such that $O^{T}O=O^{-1}O=I$ and also that $O$ has on the columns and on the rows the coordinates of the vectors of an orthonormal basis. Therefore the change of basis matrix between two orthonormal basis is indeed an orthogonal matrix.
But the columns and the rows are coordinates of the vectors of an orthonormal basis with respect to which scalar product?
Usually of course the standard scalar product is used so all the matrices that I saw were orthogonal "with respect to the standard scalar product" (if I can say that).
But is it possible to build an orthogonal matrix which has on the rows the coordinates of the vectors of an orthonormal basis with respect to a generic positive definite symmetric bilinear form $\phi$ ? Is in this case the matrix "orthogonal with respect to $ \phi$"?
This confuses me also because from the definition $O^{T}O=I$ and that $O^{T}O$ looks much like a standard scalar product, even if it should be $trace(O^{T}O)$.
Does this make sense? Could anyone make clear for me if a matrix that is "orthogonal with respect to $\phi$" can exist and if it should be consider orthogonal?
Thanks a lot for your help