Determinant identities for Laplace matrices
Let \(L\) be a real \(n\times n\) matrix. If \(A=\{a_1,\dots,a_m\}\) and \(B=\{b_1,\dots,b_m\}\) are subsets of \(\{1,\dots,n\}\) with \(a_1<\dots<a_m\), \(b_1<\dots<b_m\), denote by \(D^A_B\) the minor of \(L\) obtained by deleting the rows in~\(A\) and columns in~\(B\). Sylvester's determinantal identity states that \[ D^A_B(D^{a_k}_{b_l})^{m-2} = (-1)^{k+l}\sum_{\pi\in S_m\atop\pi(l)=k} \text{sgn}\,\pi\prod_{1\leq i\leq m\atop i\neq l}D^{a_{\pi(i)}a_k}_{b_ib_l} \] for all \(k,l=1,\dots,m\) [see, e.g., \textit{T. Muir}, A treatise on the theory of determinants. Revised and enlarged by W.~H.~Metzler, Dover Publ. Inc., New York (1960) and \textit{B.~Leclerc}, Adv. Math. 100, 101--132 (1993; Zbl 0804.05074)]. Based on this identity, the present authors show that if \(L\) is a Laplace matrix (i.e., symmetric with row and column sums zero), then \(D^A_B\) can be expressed in terms of the \(n\choose 2\) minors obtained by deleting two rows and the corresponding columns. They apply this result to a problem concerning electrical networks and to another problem on spanning tree enumeration. These problems were the original motivation of this paper.
scientific article; zbMATH DE number 5650052
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