Editing Initial condition (section)

===Discrete time===

A linear [[matrix difference equation]] of the homogeneous (having no constant term) form <math>X_{t+1}=AX_t</math> has closed form solution <math>X_t=A^tX_0</math> predicated on the vector <math>X_0</math> of initial conditions on the individual variables that are stacked into the vector; <math>X_0</math> is called the vector of initial conditions or simply the initial condition, and contains ''nk'' pieces of information, ''n'' being the dimension of the vector ''X'' and ''k''&nbsp;=&nbsp;1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable ''X''; that behavior is [[stability (mathematics)|stable]] or unstable based on the [[eigenvalue]]s of the matrix ''A'' but not based on the initial conditions.

Alternatively, a dynamic process in a single variable ''x'' having multiple time lags is 

:<math>x_t=a_1x_{t-1} +a_2x_{t-2}+\cdots +a_kx_{t-k}.</math>

Here the dimension is ''n''&nbsp;=&nbsp;1 and the order is ''k'', so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is ''nk''&nbsp;=&nbsp;''k''. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its [[Eigenvalues and eigenvectors#Dynamic equations|characteristic equation]] <math>\lambda^k-a_1\lambda^{k-1} -a_2\lambda^{k-2}-\cdots -a_{k-1}\lambda-a_k=0</math> to obtain the latter's ''k'' solutions, which are the [[characteristic value]]s <math>\lambda_1, \dots , \lambda_k,</math> for use in the solution equation

:<math>x_t=c_1\lambda _1^t+\cdots + c_k\lambda _k^t.</math>

Here the constants <math>c_1, \dots , c_k</math> are found by solving a system of ''k'' different equations based on this equation, each using one of ''k'' different values of ''t'' for which the specific initial condition <math>x_t</math> Is known.