Question

Suppose, in a TREE, every node has a weight, and consider reward of each node is a fraction from the nodes in the subtree rooted at that node. For example if node a has b, c, d in his subtree (not necessarily immediate child) and they have weights w_b, w_c, w_d, then, a's reward is [k_1*w_b + k_2*w_c + k_3*w_d]. So we can express the reward vector R = A^T W, where A is the matrix with rows denoting the fractions it gets from the other nodes and W is the weight vector. Can you comment on the singularity of \[I - A^T \] . I guess it will not be singular since we have the special tree structure o