Am I correct that any square matrix can be thought of as a directed graph? If so, how do I determine whether or not it contains cycles? Is it by looking for imaginary eigenvalues?
Yes, any square matrix can be thought of as a directed graph, particularly when you interpret the matrix as an adjacency matrix of the graph. Here's how this works and how you can determine whether the graph contains cycles:
- Adjacency Matrix: A square matrix ( A ) of size ( n \times n ) can represent a directed graph with ( n ) vertices. The entry ( A[i][j] ) indicates the presence (and possibly the weight) of a directed edge from vertex ( i ) to vertex ( j ).