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) ∈ Cm×n ×Cm×p, and let P(A,B) = {(E,F ) ∈ Cm×n ×Cm×p : there is X ∈ Cn×p such that (A− E)X = B − F}. The total least square problem concerns the determination of the existence of (E,F ) in P(A,B) having the smallest Frobenius norm. In this paper, we characterize elements of the set P(A,B) and derive a formula for ρ(A,B) = inf {‖[E|F ]‖ : (E,F ) ∈ P(A,B)} , for any unitarily invariant norm ‖·‖ on Cm×(n+p), where [E|F ] denotes them×(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 (E,F ) ∈ P(A,B) attaining ρ(A,B). The results cover those on the total least square problem, and those of Huang and Yan on the existence of (E,F ) ∈ P(A,B) so that [E|F ] has the smallest spectral norm. AMS Subject Classifications 65F20, 65F35