If A is the adjacency matrix of a digraph G, what does the (i, j) entry of AA T represent if i ≠ j
If A is the adjacency matrix of a digraph G, what does the (i, j) entry of AA T represent if i ≠ j
The (i, j) entry of the product AA^T, where A is the adjacency matrix of a directed graph G, represents the number of walks of length 2 from node i to node j in G.
Here are the steps to understand why this is true:
Let G be a directed graph with n nodes and adjacency matrix A. The (i, j) entry of A is 1 if there is a directed edge from node i to node j in G, and 0 otherwise.
The product AA^T is defined as the matrix product of A and its transpose A^T. The (i, j) entry of AA^T is given by the dot product of the ith row of A with the jth column of A^T.
To compute the dot product of the ith row of A with the jth column of A^T, we need to multiply the entries of these vectors element-wise and add up the products. That is,
(AA^T)ij = ∑k=1 to n(Aik * Akj)
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