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Let $M$ be a square $N\times N$ matrix with linearly independent columns.

I was wondering why the $QR$ decomposition is unique in this case and how to show it? The linked question below does not explain why $Q$ is also unique and I didn't quite understand its explanation of why $R$ is unique also.

Siong Thye Goh
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user376532
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    I remember that I managed to show that the $QR$ factorization was unique through the fact that the inverse of a matrix was unique. You know that the columns of $M$ are linearly independent, and hence $M$ is nonsingular and thus has an inverse.

    You are likely expected to do it via induction, but there are other ways to execute the proof.

    The way that you usually prove uniqueness is that you assume that there exists another QR factorization, perform some algebra, and show that the two distinct factorizations are actually the same.

     – Decaf-Math Oct 09 '16 at 00:12

1 Answers1

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If $Q_1R_1=Q_2R_2$ then $Q_2^{-1}Q_1=R_2R_1^{-1}$. You can check that $R_2R_1^{-1}$ is upper triangular and of course the left hand side is orthogonal. This the problem is reduced to showing that the only upper triangular orthongal matrix is the identity. This is easy to see by taking the successive inner products of the columns of the upper triangular matrix.

Rene Schipperus
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    Only upper triangular orthogonal matrix may not be identity,you can have 1 and -1 alternatively in the diagonal.So upper triangular orthogonal matrix is unique up to multiplication by a diagonal matrix with entries 1 or -1. – Soumyadip Sarkar Aug 17 '19 at 06:11