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Fundam. Prikl. Mat., 2004, Volume 10, Issue 3, Pages 245–254 (Mi fpm771)  

This article is cited in 1 scientific paper (total in 1 paper)

An interlacing theorem for matrices whose graph is a given tree

C.-M. da Fonseca

University of Coimbra

Abstract: Let $A$ and $B$ be $(n\times n)$-matrices. For an index set $S\subset\{1,\ldots,n\}$, denote by $A(S)$ the principal submatrix that lies in the rows and columns indexed by $S$. Denote by $S'$ the complement of $S$ and define $\eta(A,B)=\sum\limits_S\det A(S)\det B(S')$, where the summation is over all subsets of $\{1,\ldots,n\}$ and, by convention, $\det A(\varnothing)=\det B(\varnothing)=1$. C. R. Johnson conjectured that if $A$ and $B$ are Hermitian and $A$ is positive semidefinite, then the polynomial $\eta(\lambda A,-B)$ has only real roots. G. Rublein and R. B. Bapat proved that this is true for $n\leq3$. Bapat also proved this result for any $n$ with the condition that both $A$ and $B$ are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any $n$ under the additional assumption that both $A$ and $B$ are matrices whose graph is a tree.

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English version:
Journal of Mathematical Sciences (New York), 2006, 139:4, 6823–6830

Bibliographic databases:

UDC: 512.643

Citation: C. da Fonseca, “An interlacing theorem for matrices whose graph is a given tree”, Fundam. Prikl. Mat., 10:3 (2004), 245–254; J. Math. Sci., 139:4 (2006), 6823–6830

Citation in format AMSBIB
\Bibitem{Da 04}
\by C.~da Fonseca
\paper An interlacing theorem for matrices whose graph is a~given tree
\jour Fundam. Prikl. Mat.
\yr 2004
\vol 10
\issue 3
\pages 245--254
\mathnet{http://mi.mathnet.ru/fpm771}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2123353}
\zmath{https://zbmath.org/?q=an:1068.05017}
\transl
\jour J. Math. Sci.
\yr 2006
\vol 139
\issue 4
\pages 6823--6830
\crossref{https://doi.org/10.1007/s10958-006-0394-1}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33750510693}


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    This publication is cited in the following articles:
    1. Borcea, J, “Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products”, Duke Mathematical Journal, 143:2 (2008), 205  crossref  mathscinet  zmath  isi
  • Фундаментальная и прикладная математика
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