Extension of the total least square problem using general unitarily invariant norms

Let m, n, p be positive integers such that m ≥ n+p. Suppose (A, B) ∈ C m × n  × C m × p , and let The total least square problem concerns the determination of the existence of ( E, F ) in having the smallest Frobenius norm. In this article, we characterize elements of the set and derive a formula for for any unitarily invariant norm ‖·‖ on C m × ( n + p ), where [E|F] denotes the m× (n+p) matrix formed by the columns of E and F. Furthermore, we give a necessary and sufficient condition on ( A, B ) and the unitarily invariant norm ‖·‖ so that there exists attaining ρ (A,B). The results cover those on the total least square problem, and those of Huang and Yan on the existence of so that [E|F] has the smallest spectral norm.