# Is the the transformation of the transposed matrix equal to transpose of the transformed matrix?

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 you want $(1)$ to hold for all $A$, then $CC^T$ must be a scalar multiple of the identity matrix. If one only finds that $(1)$ holds for some $A$, I don't think we can say anything about $C$ apart from the fact that it is nonsingular. Consider $A=0$ for instance. Feb 5, 2022 at 18:30

$$\widetilde {A}^T=(DAC)^T=C^TA^TD^T$$ $$\widetilde {A^T}=DA^TC$$
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 $$D^TD=I$$.