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

Abstract

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.

Keywords:

• Unitarily invariant norms
• Range spaces
• Total least squares

Keywords:

• AMS Subject Classifications: 65F20
• 65F35

Acknowledgments

Research of the first author was partially supported by a USA NSF grant. Research of the second and third author were partially supported by NSF of Shandong Province (Y20000A04).

Notes

§ Li is an Honorary Professor of the Heilongjiang University, and an Honorary Professor of the University of Hong Kong. This article was presented by him at the Robert Thompson Matrix Theory Meeting, San Jose State University, 13 November, 2004.

Additional information

Notes on contributors

Chi-Kwong Li§

§ Li is an Honorary Professor of the Heilongjiang University, and an Honorary Professor of the University of Hong Kong. This article was presented by him at the Robert Thompson Matrix Theory Meeting, San Jose State University, 13 November, 2004.