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

Priyanka Grover, Krishna Kumar Gupta

Abstract

Some questions raised in [K. Ziętak, 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 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 $\varepsilon$-orthogonality with respect to the Ky Fan $p$-$k$ norm are also derived.

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

Abstract

Some questions raised in [K. Ziętak, 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 Ky Fan - norm is computed. A characterization for the best approximations with respect to the Ky Fan - norms is given. Further, necessary and sufficient conditions for -orthogonality with respect to the Ky Fan - norm are also derived.
Paper Structure (3 sections, 21 theorems, 84 equations)

This paper contains 3 sections, 21 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Subdifferential of the Ky Fan $p$-$k$ Norm
  3. Best approximants with respect to the Ky Fan $p$-$k$ norms

Key Result

Proposition 2.1

bhatia2013matrixbhatia2009positive Let $B\in{\mathcal{G}}(I).$ Then, for any $H\in{\mathbb H}_n({\mathbb C}),$ where $f^{(1)}(D)$ is the matrix whose $(i,j)$-entry is $f^{(1)}(\lambda_i, \lambda_j),$ and $\circ$ denotes the Schur-product of two matrices.

Theorems & Definitions (35)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3: zalinescuGeneralconvex
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Definition 2.7
  • ...and 25 more