the eigenvalues indicate the system stability, if an eigenvalue has a positive real part then the system is unstable. all eigenvalues should have negative real part so the system be stable

the eigenvectors help you find the controllability and observability of the system by Popov-Belevitch-hautus method

eigen values and corresponding eigen vectors also give an idea about the convergence of systems and filters in control perspective. The eigen values are found from the correlation matrix of input observations. So it basically tells you about how fast or slow your system will converge to an optimal value of a typical set of observations.

Thank you Elie and Prof. Amir for the reply.

Suppose we have a second order system. Now how do we relate eigen vectors and eigenvalues with the state trajectory?

(I'm editing my question to add this)

Eigen values are actually related to poles of the System. Infact Each pole of system (In Transfer Function form) is eigen value of matrix A (In State Space form). So eigen values with negative Real part indicate that poles of the system lie in Left Half plane which in turn indicates that System is stable and if System is stable all its states are stable i.e. State trajectories converge.

For a Second order system

x'=Ax+Bu, y=Cx+Du

The equivalent T/F rep of the above state space rep. is given as

T/F= C*inv(SI-A)B+D

If you look at the T/F it is clear that Poles of the system are the roots of equation

Det(SI-A)=0, which are exactly equal to eigen values of the matrix A ...

...................................................................................................................

Now if eigen values are negative, System is stable, why?

Because if Lambda1 is eigen value and x is the corresponding eigen vector then...

x'=Ax= lambda*x

x'=lambda*x, solution of this equation will be an exponentially stable "x" if "lambda" is negative, which is the state trajectory, This is the intuition that Eigen value gives regarding the state trajectories

For example for a system with "-2" as eigen value, x'=-2x will result in a state vector which will be exponentially stable

The eigenvalues are the system modes which are also poles of the transfer function in a linear time-invariant system . The eigenvectors are elementary solutions. If there is no repeated eigenvalue then there is a basis for which the state-trajectory solution is a linear combination of eigenvectors. If there are repeated eigenvalues then the solution is a linear combination of generalized eigenvectors in the sense that each eigenvalue has associated a generalized eigensolution which is a combination of elementary solutions equal to the multiplicity of the eigenvalue in the minimal polynomial of the matrix describing the system dynamics.

Solve a second order numerical example and analyze. then you will understand clearly.

Ok. So I'm gonna give a different perspective than the others already gave (specially because you are asking about their relations to the state trajectories).

So qualitatively speaking (for more information, you could check out Khalil's book on Nonlinear Systems, specifically the second chapter) you can think of the eigenvectors as defining directions and the eigenvalues the behavior of a linear system in those directions. That is, if you have a strictly real eigenvalue, the system will either exponentially decrease or increase in the corresponding eigenvector direction (it will decrease if the eigenvalue is negative or increase if it is positive). If you have a pair of complex conjugate eigenvalues, the system will oscillate around the directions given by the corresponding eigenvectors while either exponentially decreasing towards the origin, or increasing away from it according to the signal of the eigenvalues real part. In addition, we can also tell how fast the system oscillates in this direction by looking at the magnitude of the imaginary part of the eigenvalues.

Putting Victor's reply in a slightly different way, if you consider the system as the initial value problem:

dx(t)/dt = A x(t), x(0)=x_0

Setting the initial condition, x(0), to be equal to an eigenvalue direction, then the resulting trajectory will excite only the mode associated with the corresponding eigenvalue. Thus, for a 2nd order linear system with 2 distinct real eigenvalues, setting x(0) to be equal to an eigenvector means that the resulting trajectory will evolve only in the direction of the eigenvector (towards the origin if the eigenvalue is negative, and away from it if it is positive) and will evolve exponentially with the time constant of only the associated eigenvalue. i,e, the trajectory will lie on the straight line that passes through the origin and the eigenvector.

Eigenvalues indicates to the stability of the system ,if the real part is negative then the system is stable but if the real part of the eigenvalue is positive then the system is unstable .

on other hand , eigenvalue comes into play representing the magnitude of the stretching of matrix A and eigenvector comes into play representing the direction of the stretching of matrix A .

This equation (A*x=lambda*x ) represents the relationship between matrix A and the eigenvector matrix x and the eigenvalue lambda .