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

An interlacing theorem for matrices whose graph is a given tree

C. M. da Fonseca


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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' the complement of S and define h(A,B) = åS 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