Anti-adjacency matrix is a way to represent a directed graph as a square matrix, whose entries show whether there is a directed edge from a vertex to another one. This paper focuses on the properties of anti-adjacency matrix of windmill graph K(4,n), such as its characteristic polynomial and eigenvalues. The general form of characteristic polynomial is established by analyzing the degree of vertices and edges, and the cyclic induced subgraphs. Furthermore, the eigenvalues of a windmill graph K(4,n) and its multiplicity are derived from the general form of its characteristic polynomial.