Thin QR factorization from Cholesky decomposition ($A=QR,Q^{T}Q=I,R^{T}R=A^{T}A$)

If a real $m$-by-$n$ matrix $A$ has full column rank, with $m\ge n$, then $A^{T}A$ is positive-definite. Its upper-triangular Cholesky decomposition $A^{T}A=R^{T}R$, with positive diagonal, is unique. Defining $Q=A{R}^{-1}$ gives $Q^{T}Q=I$ and $A=QR$. Any other such factorization yields another Cholesky factor of $A^{T}A$, so its $R$ and then its $Q$ must agree. This proves existence and uniqueness; it does not assert that forming $A^{T}A$ is the most numerically stable algorithm.