Final answer:
To prove the uniqueness of the reduced non-negative QR factorization, one should consider the uniqueness of the nonnegative Cholesky factorization of the Hermitian matrix AᵣA.
Step-by-step explanation:
To prove that the reduced non-negative QR factorization is unique given that A∈Cᵍ¹⁴⁹,m≥n,rank(A)=n, we first consider the matrix AᵣA, where Aᵣ denotes the Hermitian transpose of A. The nonnegative Cholesky factorization theorem ensures that if AᵣA is Hermitian and positive definite, then there exists a unique nonnegative lower triangular matrix L such that AᵣA = LLᵣ. It is given that A has full rank n, which implies that AᵣA is indeed positive definite and therefore, the uniqueness of the nonnegative Cholesky factorization of AᵣA follows.
Next, let's look at the QR factorization where A = QR with Q being an m x n unitary matrix and R an n x n upper triangular matrix. Because A has full rank n, R also has full rank and is thus invertible. The original matrix A can be uniquely determined by Q and the nonnegative diagonal elements of R, making the QR factorization unique. Finally, this uniqueness of QR translates to the uniqueness of the reduced non-negative QR factorization of A.