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 and be ´
n)-matrices.
For an index set Ì {1,¼,n},
denote by the principal
submatrix that lies in the rows and columns indexed
by .
Denote by
the complement of and define h(A,B) = åS det A(S)
det B(S'), where the summation
is over all subsets of ¼,n} and, by
convention, Æ) = det B(Æ) =
1.
C. R. Johnson conjectured that if and are Hermitian and
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
£
3.
Bapat also proved this result for any with the condition that
both and 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
under the additional assumption that both and are matrices whose graph
is a tree.
Location: http://mech.math.msu.su/~fpm/eng/k04/k043/k04313h.htm
Last modified: February 28, 2005