Uniqueness of the QR-factorization I'm preparing a talk about the QR-factorization.
I alredy have proved the existence of it:
If $A \in \mathbb{R}^{n \times m} $ there exists an orthogonal matrix  $Q \in \mathbb{R}^{n \times n}$ and an upper triangular $R \in \mathbb{R}^{n \times m}$ with $A=QR$.
There is a mistake in the following theorem, see my answer below.
Now I want to prove the following theorem:
If $A \in \mathbb{R}^{n \times m} $ with $rank(A)=m$ has the QR-factorization $A=QR$ where $R$ has positive diagonal entries, then the Q and R are unique.
But all literature use proves with another factorisation. So i dont now how to handle, maybe sombody can help, thanks.

Once you have one $QR$ factorization, say $A=Q_1R_1$, then it is easy to produce another one by defining $Q_2=Q_1B$ and $R_2=B^{-1}R_1$. But for $Q_2$ and $R_2$ to be orthogonal and upper triangular, respectively, $B$ must be orthogonal and diagonal. That means it can only have $\pm 1$ as elements on the diagonal. If $R_1$ already has positive diagonal entries, then $B$ must be the identity.
