Identities for minors of the Laplacian, resistance and distance matrices
نویسندگان
چکیده
منابع مشابه
Identities for minors of the Laplacian, resistance and distance matrices
It is shown that if L and D are the Laplacian matrix and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, upto a sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is ...
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Proof. We first recall that every non-singular matrix B can be written B = QR, where Q is an orthonormal matrix Q and R is upper-triangular matrix R with positive diagonals1 We will use a slight variation of this fact, writing B = RQ. Now, since QT = Q−1, QAQT has exactly the same eigenvalues as A. Let Rt be the matrix t ∗R+ (1− t)I, and consider the family of matrices Mt = RtQAQR t , as t goes...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2011
ISSN: 0024-3795
DOI: 10.1016/j.laa.2011.03.028