Discrete Combat Models y(n) = the number of combatants in the Y-force after period n. The future state is then x(n+1) and y(n+1), respectively. Thus, we have using our paradigm, x(n+1) = x(n) + Change y(n+1)=y(n) + Change. Figure 1 provides the information that reflects change. Our dynamical system of equations is: x(n+1) = x(n) ­k 1 y(n) y(n+1) = y(n) ­k 2 x(n) We define our starting conditions as the size of the combatant forces at time period zero: x(0)=x 0 and y(0)=y 0 . x 1 - k 1 X n , X 0 = 0 X n + 1 = y 0 - k 2 1 (2) We will use the matrix solution method using eigenvalues and eigenvectors to find the analytical solution of equation (2). Further, we will charac- terize the solution we found in terms of only k 1 , k 2 , x 0 , and y 0 . This is significant as we can quickly write the solution to this form of Lanchester's equation modeled as in equation (2). Let's begin by defining eigenvectors and eigen- values: Let A be a n × n matrix. The real number is called an eigenvalue of A if there exists a nonzero vector x in R n such that Ax = x . (3) The nonzero vector x is called an eigenvector of