I initially thought that the answer was true. Since a matrix has unique QR factorization if it is invertible. And we are given that columns are linearly independent which ensures that matrix $A$ is invertible.However, other sources are telling me otherwise. Is there a possible counterexample that can prove it false? Thank you
2026-04-05 17:47:50.1775411270
There is a unique QR factorization of any matrix $A$ that has full column rank — true or false?
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If $Q_1R_1=Q_2R_2$ where $Q_i$ are orthogonal and $R_i$ are upper triangular, then $Q=Q_2^{-1}Q_1=R_2R_1^{-1}$ (assuming $A$ is invertible). So the question boils down to whether an orthogonal matrix can be upper triangular.
The answer is yes: any diagonal matrix consisting of $\pm1$. So $QR$ decompositions are unique only up to multiplication by a diagonal matrix of $\pm1$s, unless they are restricted in some way (usually by requiring that $R$ has positive main diagonal).