## 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 x^{2}_{1} + ⋯ + x^{2}_{m-p} - x^{2}_{m - p + 1} - ⋯ - x^{2}_{m} = 1. By contrast, the familiar results are associated with the sphere x^{2}_{1} + ⋯ + x^{2}_{m} = 1.

Original language | English (US) |
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Pages (from-to) | 115-130 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 70 |

Issue number | C |

DOIs | |

State | Published - Oct 1985 |

## All Science Journal Classification (ASJC) codes

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