I'm self studying linear algebra and I got stuck with one doubt.
I've a transformation $T$ represented by an orthogonal matrix $A$ , so $A^TA=I$. This transformation leaves norm unchanged.
I do a basis change using a matrix $B$ which isn't orthogonal , then the form of the transformation changes to $B^{-1}AB$ in the new basis( A similarity transformation).
Since we only changed our representation of the transformation $T$ then transformation $B^{-1}AB$ should also leave norm unchanged which means that $B^{-1}AB$ should be orthogonal.
Therefore $B^{-1}AB$.${{[B^{-1}AB}}]^T=I$.
This suggests that $B^TB=I$ which means it is orthogonal, but that is a contradiction.
Can anyone tell me if what I did wrong. Thank you.
A transformation $\mathscr{A}$ is orthogonal iff its matrix representation is orthogonal with respect to an standard orthogonal basis. And the transition matrix between two standard orthogonal bases must be orthogonal.
$B_1=\{e_1,\dots,e_n\}$, $B_2=\{f_1,\dots,f_n\}$ are two standard orthogonal bases. $A_1$ and $A_2$ are representations of $\mathscr{A}$ with respect to $B_1,B_2$. Then $A_1,A_2$ are orthogonal matrices. $$ \mathscr{A}(e_i)=\text{the $i$th colume of $A_1$};\;\;\mathscr{A}(f_i)=\text{the $i$th colume of $A_2$} $$ If $P$ is the transition matrix between $B_1$ and $B_2$( that is $(e_1,\dots,e_n)=(f_1,\dots,f_n)P$), then $P$ is orthogonal.