Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s. Let D_ij denote the sum of all the weights lying in the path connecting the vertices i and j of T. We now say that D_ij is the distance between i and j. Define D ≔ [D_ij], where D_ii is the s × s null matrix and for i ≠ j, D_ij is the distance between i and j. Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of order s. If i and j are adjacent, then define L_ij ≔ − W⁻¹_ij , where W_ij is the weight of the edge (i, j). Define \({L_{ii}}: = \sum\limits_{i \ne j,j = 1}^n {W_{ij}^{ - 1}}.\) The Laplacian of G is now the ns × ns block matrix L ≔ [L_ij]. In this paper, we first note that D⁻¹ − L is always nonsingular and then we prove that D and its perturbation (D⁻¹ − L)⁻¹ have many interesting properties in common.

令T为具有n个顶点的树。我们为T的每个边缘分配一个权重，该权重是某个固定阶数的正定矩阵，例如s。让d _IJ表示所有躺在路连接顶点的权重的总和我和Ĵ的牛逼。现在我们说D _ij是i和j之间的距离。定义d ≔[ d _IJ ]，其中d _II是小号×小号零矩阵和用于我≠ Ĵ，d_ij是i和j之间的距离。令G为具有n个顶点的任意连通加权图，其中每个权重为阶s的正定矩阵。如果i和j相邻，则定义L _ij -W ^-1 _ij ，其中W _ij是边缘（ i，j）的权重。定义\（{L_ {ii}}：= \ sum \ limits_ {i \ ne j，j = 1} ^ n {W_ {ij} ^ {-1}}。\）G的拉普拉斯算子现在是ns× ns块矩阵L [ L_ij ]。在本文中，我们首先要注意d ^-1 -大号总是奇异的，然后我们证明了d及其扰动（ d ^-1 -大号） ^-1有许多共同的有趣的性质。