02203299

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02203299 da Fonseca, C.M. An interlacing theorem for matrices whose graph is a given tree. Fundam. Prikl. Mat. 10, No. 3, 245-254 (2004); translation in J. Math. Sci., New York 139, No. 4, 6823-6830 (2006). 2004 MGU, Tsentr Novykh Informatsionnykh Tekhnologij, Moskva; Izdatel'skij Dom "Otkrytye Sistemy", Moskva RU EN Hermitian matrices eigenvalues weighted graphs Let $A, B\in M_n({\Bbb C})$ be square $n\times n$ matrices with complex entries. For a subset $S$ of $N:= \{1,\ldots, n\}$ let $A(S)$ be a submatrix of $A$ that lies in the intersection of rows and columns indexed by the elements of $S$. Let $S'=\{1,\dots,n\} \setminus S$ and $$\eta(A,B)=\sum_{S\subseteq N} \det A(S)\det B(S'),$$ here it is assumed that $\det A(\emptyset) =\det B(\emptyset)=1$. The celebrated Jonson conjecture says that if both $A$ and $B$ are Hermitian and $A$ is also positive semidefinite then the polynomial $\eta(\lambda A,-B)$ has only real roots $\lambda_l^A(B)$, $l=1,\dots, n$. This conjecture was approved for $n\le 3$ and for arbitrary $n$ and tridiagonal matrices $A,B$. Also some natural interlacing conjecture was posed which is generalized by the author in the following way: Let $S\subseteq N$, $\vert S\vert =k$. If both $A$ and $B$ are Hermitian and $A>0$ then the roots $\lambda_l^{A_S}(B_S)$ of $\eta(\lambda A(S'),-B(S'))$ interlace with $\lambda_l^A(B)$ as follows: $\lambda_l^A(B)\le \lambda_l^{A_S}(B_S)\le\lambda_{l+k}^A(B)$. In the paper this problem for matrices whose graph is a tree is worked out. The author proves that if $A,B$ are the matrices whose graph is a fixed tree and $A>0$ then $\eta(\lambda A, -B)$ has real roots, say $\lambda_1\le \lambda_2\le \cdots\le \lambda_n$. Moreover, if $\mu_1\le\cdots\le \mu_{n-k}$ are the roots of $\eta(\mu A(S'),-B(S'))$, where $S\subseteq N$, $\vert S\vert =k$, then $\lambda_l\le\mu_l\le\lambda_{l+k}$, $l=1,\dots, l-k$. A series of useful examples and corollaries are presented. Alexander E. Guterman (Moskva)