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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,

$$\widetilde {A}^T\stackrel{?}{=}\widetilde {A^T} \tag{1}$$

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  • $\begingroup$ 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. $\endgroup$
    – user1551
    Feb 5, 2022 at 18:30

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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 $D^TD=I$.

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