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Incidence Matrix [Aa]

The incidence matrix symbolically describes a network. It also facilitates the testing and identification of the independent variables. Incidence matrix is a matrix which represents a graph uniquely.

For a given graph with n nodes and b branches, the complete incidence matrix Aa is a rectangular matrix of order n × b, whose elements have the following values.
Number of columns in [Aa] = Number of branches = b
Number of rows in [Aa] = Number of nodes = n
Aij = 1, if branch j is associated with node i and oriented away from node i.
= –1, if branch j is associated with node i and oriented towards node i.
= 0, if branch j is not associated with node i.
This matrix tells us which branches are incident at which nodes and what are the orientations relative to the nodes.

Properties of Complete Incidence Matrix

  1. The sum of the entries in any column is zero.
  2. The determinant of the incidence matrix of a closed loop is zero.
  3. The rank of incidence matrix of a connected graph is (n–1).

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