Properties of best approximations with respect to the Ky Fan $p$-$k$ norm, and the strict spectral approximant of a matrix

TL;DR

This paper investigates properties of best matrix approximations under the Ky Fan p-k norm, including subdifferential computation, characterization of optimal approximations, and orthogonality conditions.

Contribution

It provides new characterizations and conditions for best approximations and orthogonality in the context of the Ky Fan p-k norm, extending previous spectral approximation results.

Findings

Computed the subdifferential set of the Ky Fan p-k norm.

Provided a characterization for best approximations under this norm.

Derived necessary and sufficient conditions for ε-Birkhoff orthogonality.

Abstract

Some questions raised in [K. Zi\k{e}tak, {\it From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix}, Warsaw : Banach Center Publ., 112 Polish Acad. Sci. Inst. Math. (2017)] are discussed. To do so, the subdifferential set of the Ky Fan $p$-$k$ norm is computed. A characterization for the best approximations with respect to the Ky Fan $p$-$k$ norms is given. Further, necessary and sufficient conditions for $\varepsilon$-Birkhoff orthogonality with respect to the Ky Fan $p$-$k$ norm are also derived.