Proof first lets examine the invertible case

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Unformatted text preview: R and Q are unique. Proof. First, lets examine the invertible case. Uniqueness: Let A = Q1 R1 = Q2 R2 with Q1 , Q2 ∈ Mn unitary and R1 , R2 upper triangular with non-negative (1) (2) diagonal entries rj,j ≥ 0 and rj,j ≥ 0. The entries on the diagonal cannot be zero because then (1) (2) 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 2 We see that M is unitary and u...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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