Linear dynamical system

Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.
Introduction.
In a linear dynamical system, the variation of a state vector 
(an formula_1-dimensional vector denoted formula_2) equals a constant matrix
(denoted formula_3) multiplied by 
formula_2. This variation can take two forms: either 
as a flow, in which formula_2 varies 
continuously with time
or as a mapping, in which 
formula_2 varies in discrete steps
These equations are linear in the following sense: if 
formula_9 and formula_10 
are two valid solutions, then so is any linear combination 
of the two solutions, e.g., 
formula_11 
where formula_12 and formula_13
are any two scalars. The matrix formula_3 
need not be symmetric.
Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.
Solution of linear dynamical systems.
If the initial vector formula_15
is aligned with a right eigenvector formula_16 of 
the matrix formula_3, the dynamics are simple
where formula_19 is the corresponding eigenvalue;
the solution of this equation is 
as may be confirmed by substitution.
If formula_3 is diagonalizable, then any vector in an formula_1-dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted formula_23) of the matrix formula_3.
Therefore, the general solution for formula_9 is 
a linear combination of the individual solutions for the right
eigenvectors
Similar considerations apply to the discrete mappings.
Classification in two dimensions.
The roots of the characteristic polynomial det(A - λI) are the eigenvalues of A. The sign and relation of these roots, formula_28, to each other may be used to determine the stability of the dynamical system 
For a 2-dimensional system, the characteristic polynomial is of the form formula_30 where formula_31 is the trace and formula_32 is the determinant of A. Thus the two roots are in the form:
and formula_35 and formula_36. Thus if formula_37 then the eigenvalues are of opposite sign, and the fixed point is a saddle. If formula_38 then the eigenvalues are of the same sign. Therefore, if formula_39 both are positive and the point is unstable, and if formula_40 then both are negative and the point is stable. The discriminant will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).