Let $A$ be a square matrix expressed in a basis $\{\mathbf e_i\}$ and let $C$ be the transition matrix from $\{\mathbf e_i\}$ to a new basis $ \{\tilde{\mathbf e}_i\}$, so $\widetilde {A}=DAC$, with $D=C^{-1}$. This question has come up for me: is the transpose of a transformed matrix equal to the transformation of the transposed matrix? That is,
$$\widetilde {A}^T\stackrel{?}{=}\widetilde {A^T} \tag{1}$$
If we write the corresponding expressions in terms of the old basis,
$$\widetilde {A}^T=(DAC)^T=C^TA^TD^T$$ $$\widetilde {A^T}=DA^TC$$
And we see that $\widetilde {A}^T=\widetilde {A^T}$ if
$$D^{-1}C^TA^TD^TC^{-1}=A^T$$
$$CC^TA^TD^TD=A^T$$
So this only happens in the case of orthogonal transformations so that $CC^T=I$ and $D^TD=I$.