FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2004, VOLUME 10, NUMBER 3, PAGES 245-254

An interlacing theorem for matrices whose graph is a given tree

C. M. da Fonseca

Abstract

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Let $A$ and $B$ be $(n$´ n)-matrices. For an index set $S$Ì {1,¼,n}, denote by $A(S)$ the principal submatrix that lies in the rows and columns indexed by $S$. Denote by $S\text{'}$ the complement of $S$ and define $$h(A,B) = å_S det A(S) det B(S'), where the summation is over all subsets of $\{1,$¼,n} and, by convention, $det\; A($Æ) = det B(Æ) = 1. C. R. Johnson conjectured that if $A$ and $B$ are Hermitian and $A$ is positive semidefinite, then the polynomial $$h(lA, -B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for $n$£ 3. 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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Last modified: February 28, 2005