POSITIVE-DEFINITE MATRICES AND THEIR APPLICATIONS TO THE PROBLEMS OF INTERNATIONAL MATHEMATICAL OLYMPIADS

Nabikhonov Nabikhon Yokubjon ugli; Aytjanova Gulayym Torabevna

Authors

Nabikhonov Nabikhon Yokubjon ugli Student of the National University of Uzbekistan, winner of the International Olympiads Author
Aytjanova Gulayym Torabevna Student of the National University of Uzbekistan, winner of the International Olympiads Author

Keywords:

symmetric real matrix, Hermitan matrix, positive-definite, positive-semidefinite, negative-definite, negative-semidefinite, Gauss’ Theorem, Gram matrix, Euclidean space, diagonalizable, skew-symmetric matrix, eigenvalues

Abstract

This article reflected on positive and negative defined matrices. Suitable definitions were given for such matrices. Theorem about the general appearance positive-definite matrices is given with its proof. The connection between a Gram Matrix and a positively defined matrix is also given. Solutions to complex Olympic problems are given. At the end of the article, enough problems are given to work independently.

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