Uniqueness of Thin QR Factorization.

Question

Let $A \in \mathbb C^{m \times n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m \times n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal elements, then $Q$ and $R$ are unique.

$Q^*$ is tranpose of $Q$

Is a exercise of Numerical Matrix Analysis of Ipsen

How do I can prove this? I need help

$A=Q_1R_1=Q_2R_2$ then $R_1=Q_1^*Q_2R_2$, so I need to prove that $Q_1^*Q_2=I_n$

score 7 · Answer 1 · answered Oct 24 '14 at 21:29

7

Let $A=Q_1R_1=Q_2R_2$ then we have $Q_2^{-1}Q_1=R_2 R_1^{-1}$ but $Q_2^{-1}Q_1$ is unitary matrix so it's diagonalizable and their eigenvalues belong to the unit circle of the complex plane and it's a triangular matrix with positive diagonal elements (eigenvalues) so these eigenvalues are equal to $1$ and then this matrix is similar to $I_n$ hence it's equal to $I_n$. Conclude.

answered Oct 24 '14 at 21:29

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$Q_1$ and $Q_2$ don't have inverses. They're not square. (This is the reduced $QR$ factorization OP is asking about.) – Eric Auld Jan 29 '18 at 23:43
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@Eric Auld it still works with $Q_2^*$ instead of $Q_2^{-1}$. – Algebraic Pavel Nov 22 '19 at 14:05

Uniqueness of Thin QR Factorization.

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