Given any two ellipses in $\mathbb{R}^2$, does there exist a linear (but not necessarily orthogonal) change of coordinates $(x_0 = ax+by +k, y_0 = cx+dy +l; ad−bc \neq 0)$ so that both the ellipses have axes parallel to the new coordinate axes - namely, under the new coordinate system?
I think yes, but am unable to image how. If it were one ellipse, then easily the transformation could have been designed. But, in the question, there are two different ellipses. I think we must map one ellipse to another and then use compose it with the change of coordinates. Is there a matrix like representation for such operations? Thanks beforehand.