Uniqueness of the QR factorization

• The previous sections have shown that the QR factorization exists.
• Let’s assume that $A$ is full column rank, then we will prove that its QR factorization is unique if we require that $r_{ii}>0$.

Proof: consider $A^{T}A$. Since $A$ is full column rank, this matrix is SPD. From $A=QR$, we get:

$$A^{T}A=R^T{Q}^{T}QR=R^{T}R$$

Denote by $L=R^T$. $L$ is a lower triangular matrix with positive entries on the diagonal. We have $A^{T}A=L{L}^T$. So $L{L}^T$ is the Cholesky factorization of $A$. This factorization is unique. So the factor $R$ is unique. But we also have:

$$Q=A{R}^{-1}$$

so the factor $Q$ is also unique. $\square$

Symmetric Positive Definite Matrices, Existence of the Cholesky factorization, QR factorization, QR using Householder transformations, QR using Givens transformations, Gram-Schmidt