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Unformatted text preview: R and Q are unique.
Proof. First, lets examine the invertible case.
Let A = Q1 R1 = Q2 R2 with Q1 , Q2 ∈ Mn unitary and R1 , R2 upper triangular with non-negative
diagonal entries rj,j ≥ 0 and rj,j ≥ 0. The entries on the diagonal cannot be zero because then
we would have an eigenvalue of 0 and then A would not be invertible. So rj,j > 0 and rj,j > 0.
We transform the identity to:
Q 1 = Q 2 R2 R1 1 Which implies that
M = Q ∗ Q 1 = Q ∗ ( Q 2 R2 R1 1 ) = R2 R1 1
2 We see that M is unitary and u...
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