Moore-penrose inverse of the incidence matrix of a tree
Moore-penrose inverse of the incidence matrix of a tree
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DOI：
10.1080/03081089708818496
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发表时间：
1997
影响因子：
1.1
通讯作者：
R. Bapat
中科院分区：
文献类型：
--
作者：
R. Bapat
设T是一棵有n个顶点的树，其中每条边都有一个方向，Q是它的顶点-边关联矩阵。证明Q的Moore-Penrose逆是如下得到的（n−1）× n矩阵M。M的行和列分别由T的边和顶点索引。如果e，ν分别是T的一条边和一个顶点，则M的（e，ν）-项是T\e中不包含ν的连通分支的顶点数，直到符号为止。此外，项的符号是正的还是负的，取决于e是朝向还是远离ν。这个结果然后被用来获得任意有向图的关联矩阵的Moore-Penrose逆的表达式。最近的一个结果，由于月球也衍生的结果。
Let T be a tree with n vertices, where each edge is given an orientation, and let Q be its vertex-edge incidence matrix. It is shown that the Moore-Penrose inverse of Q is the (n−1)× n matrix M obtained as follows. The rows and the columns of M are indexed by the edges and the vertices of T respectively. If e,ν are an edge and a vertex of T respectively, then the (e,ν)-entry of M is, upto a sign, the number of vertices in the connected component of T\e which does not contain ν. Furthermore, the sign of the entry is positive or negative, depending on whether e is oriented away from or towards ν. This result is then used to obtain an expression for the Moore-Penrose inverse of the incidence matrix of an arbitrary directed graph. A recent result due to Moon is also derived as a consequence.