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 $\hat{B}$.