Let $A \in \mathbb C^{m \times n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m \times n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal
elements, then $Q$ and $R$ are unique.
$Q^*$ is tranpose of $Q$
Is a exercise of Numerical Matrix Analysis of Ipsen
How do I can prove this? I need help
$A=Q_1R_1=Q_2R_2$ then $R_1=Q_1^*Q_2R_2$, so I need to prove that $Q_1^*Q_2=I_n$
Best Answer
Let $A=Q_1R_1=Q_2R_2$ then we have $Q_2^{-1}Q_1=R_2 R_1^{-1}$ but $Q_2^{-1}Q_1$ is unitary matrix so it's diagonalizable and their eigenvalues belong to the unit circle of the complex plane and it's a triangular matrix with positive diagonal elements (eigenvalues) so these eigenvalues are equal to $1$ and then this matrix is similar to $I_n$ hence it's equal to $I_n$. Conclude.