While trying to prove that the left-inverse of $A$ provided by the least-squares solution to $y=Ax$ has the smallest Frobenius norm, I am stuck at a point which I describe below:
Let $B$ be any left-inverse of a full-rank tall matrix $A$, i.e., $BA=I$. Let the QR-decomposition: $A=QR$. In this case, $R$ is invertible since A is full-rank and $Q$ has orthonormal columns as always.
I want to show that $\|B\|_F \ge \|BQ\|_F$. Any ideas? The rest of the proof to show that the least-squares left-inverse has the smallest Frobenius norm is in place and I will be done if I can show this.