Some Further Aspects of the Owner-Ordering Antitonicity of the Moore-Penrose Inverse

Kenneth Nordstrom

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz’s theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.

Original languageEnglish
Pages (from-to)4471-4489
Number of pages19
JournalCommunications in Statistics - Theory and Methods
Volume18
Issue number12
DOIs
Publication statusPublished - 1989
Externally publishedYes

Fingerprint

Moore-Penrose Inverse
Non-negative
Inverse function
Schur Complement
Partial ordering
Inverse matrix
Positive definite matrix
Matrix Function
Theorem
Quadratic form
Statistics

Keywords

  • comparison of vector-valued statistics
  • concentration
  • concentration ellipsoid
  • ellipsoidal cylinder
  • extremal representation of quadratic form
  • inertia
  • linear support
  • maximum
  • minimum dispersion
  • polar set
  • Schur complement

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

Some Further Aspects of the Owner-Ordering Antitonicity of the Moore-Penrose Inverse. / Nordstrom, Kenneth.

In: Communications in Statistics - Theory and Methods, Vol. 18, No. 12, 1989, p. 4471-4489.

Research output: Contribution to journalArticle

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