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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.