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 Zh. Vychisl. Mat. Mat. Fiz., 2020, Volume 60, Number 1, Pages 109–115 (Mi zvmmf11019)

Symmetric matrices whose entries are linear functions

A. V. Seliverstov

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, 127051 Russia

Abstract: There exists a large set of real symmetric matrices whose entries are linear functions in several variables such that each matrix in this set is definite at some point, that is, the matrix is definite after substituting some numbers for variables. In particular, this property holds for almost all such matrices of order two with entries depending on two variables. The same property holds for almost all matrices of order two with entries depending on a larger number of variables when this number exceeds the order of the matrix. Some examples are discussed in detail. Some asymmetric matrices are also considered. In particular, for almost every matrix whose entries are linear functions in several variables, the determinant of the matrix is positive at some point and negative at another point.

Key words: linear algebra, symmetric matrix, semidefinite programming, Hessian matrix.

DOI: https://doi.org/10.31857/S0044466920010147

English version:
Computational Mathematics and Mathematical Physics, 2020, 60:1, 102–108

Bibliographic databases:

UDC: 512.643
Revised: 26.07.2019
Accepted:18.09.2019

Citation: A. V. Seliverstov, “Symmetric matrices whose entries are linear functions”, Zh. Vychisl. Mat. Mat. Fiz., 60:1 (2020), 109–115; Comput. Math. Math. Phys., 60:1 (2020), 102–108

Citation in format AMSBIB
\Bibitem{Sel20} \by A.~V.~Seliverstov \paper Symmetric matrices whose entries are linear functions \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2020 \vol 60 \issue 1 \pages 109--115 \mathnet{http://mi.mathnet.ru/zvmmf11019} \crossref{https://doi.org/10.31857/S0044466920010147} \elib{https://elibrary.ru/item.asp?id=41806923 } \transl \jour Comput. Math. Math. Phys. \yr 2020 \vol 60 \issue 1 \pages 102--108 \crossref{https://doi.org/10.1134/S0965542520010121} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000521749800010} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85082573301} 

• http://mi.mathnet.ru/eng/zvmmf11019
• http://mi.mathnet.ru/eng/zvmmf/v60/i1/p109

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Seliverstov, “Dvoichnye resheniya dlya bolshikh sistem lineinykh uravnenii”, PDM, 2021, no. 52, 5–15