The algebra of hyperboloids of revolution

Javier F. Cabrera, Geoffrey S. Watson

Research output: Contribution to journalArticlepeer-review

Abstract

If we change the sign of p ≤ m columns (or rows) of an m × m positive definite symmetric matrix A, the resultant matrix B has p negative eigenvalues. We give systems of inequalities for the eigenvalues of B and of the matrix obtained from B by deleting one row and column. To obtain these, we first develop characterizations of the eigenvalues of B which are analogous to the minimum-maximum properties of the eigenvalues of a symmetric A, i.e. the Courant-Fischer theorem. These results arose from studying probability distributions on the hyperboloid of revolution x21 + ⋯ + x2m-p - x2m - p + 1 - ⋯ - x2m = 1. By contrast, the familiar results are associated with the sphere x21 + ⋯ + x2m = 1.

Original languageEnglish (US)
Pages (from-to)115-130
Number of pages16
JournalLinear Algebra and Its Applications
Volume70
Issue numberC
DOIs
StatePublished - Oct 1985

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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