Do QR and RQ have the same eigenvalues

Question

Let $A=QR$ where $Q\in$ $M_n(\mathbb R)$ orthogonal and $R\in M_n(\mathbb R)$ is upper triangular matrix. Do $A$ and $RQ$ have the same eigenvalues?

$\det(A-\lambda I)=\det(QR-\lambda I)=\det(QR-\lambda QQ^T)=\det Q\det(R-\lambda Q^T)=\det Q\det(RI-\lambda Q^T)=\det Q\det(RQQ^T-\lambda Q^T)=\det Q\det(RQ-\lambda I)\det Q^T=\det(RQ-\lambda I),$

so they have the same eigenvalue, is this ok?

Answer 1

Both matrices are similar: $$ A = QR = Q(RQ)Q^{-1}, $$ hence they have the same Jordan form.