It follows that we can define a new matrix $$ \hat{A}\hat{Q} = \hat{Q}\hat{U}\hat{Q}, $$ and multiply from the left with \( \hat{Q}^{-1} \) we get $$ \hat{Q}^{-1}\hat{A}\hat{Q} = \hat{B}=\hat{U}\hat{Q}, $$ where the matrix \( \hat{B} \) is a similarity transformation of \( \hat{A} \) and has the same eigenvalues as \( \hat{B} \).