Perhaps the best way to visualize eigenvalues or eigenvectors is to envision a  3-D crystal lattice structure.

Now input a planar wave into the crystal exactly in the direction of a 2-D lattice plane within the crystal: the wave will transfer to the other side of the crystal as is, with no leakage along other directions: this is the eigenvectors.

Should you however apply the wave in a direction not parallel to any one of the natural directions of the lattice - the lattice planes - then the wave will break up into components and will output in several separate directions

I hope this helps?

Hello Mr. Chris thank you for your reply, could you explain in terms of control theory system matrix,  it is the place where I had confusion , from my readings I understood that if any matrix is operated onto the eigenvector it just scales it but not change its properties , so we use this eigenvector matrix in similarity transformation to transform system matrix to diagonal matrix which implies it is decoupled

But this I am not able to understand physically

Thanks in advance

Not sure I quite understand where your confusion lies - of course if the matrix is operated onto the eigenvector it scales it but does not alter its properties - which is quite different from attempting to diagonalize a matrix though - not all matrices are diagonalizable.

Generally speaking, a matrix of order N is only diagonalizable if it has N eigenvalues

Ya if it has n linear independent vectors

Eigenvalues and eigenvectors are a mathematical concept and the physical meaning depends on the system you are investigated. It is usually related to linear problems. The typical example is the Shroedinger equation where Eigenvalues are the energy of a given eigenstate which square of the wave function is the probability density. However, e-values and e-vectors are applied to linearized problems like in Euler equations for fluid dynamics, where the eigenvectors are the Riemman variables, that represents quantities propagating as constant along the characteristic curve. 

An important application is in problems like dy/dt=My. In this case eigenvalues determine the relaxation time of the associated eigenvector. if your initial value is decomposed in eigenvectors as

y0=sum a_iv_i

the time evolution is

y(t)=sum a_i v_i exp(L_i*t)

In kinetic process you can have a null eigenvalue namely L_0 and therefore, v_0 is the stationary solution. 

The problem arise when some of the eigenvalues are positive, therefore your system is unstable, with a diverging solution, on the other hand, if all the other eigenvalues are negative, the stationary solution is given by a_0v_0. Here for sake of simplicity I have considered all eigenvalues to be simple, i.e. with a single eigenvector associated. 

The answer depends on the nature of the matrix.

If the matrix is a self adjoint linear operator on a vector space, that is, an endomorphism, then an eigenvector is a vector of the space whose direction is left unchanged by the transformation. Its length, however, may be scaled. The scaling factor is the corresponding eigenvalue. 

If your matrix is a symmetric covariant rank two tensor, you need to raise one index with the metric tensor to obtain the associated endomorphism. The proceed as above. The eigenvectors then represent the basis in which the quadric represented by the tensor has its simplest representation. The eigenvalues are then the inverse of the root of the distances from the origin to the intersection of the surface with the principal axes.

If your space does not have a metric, and there is no symmetric positive definite quadratic form to play the role of a metric, the concept of an eigenvector cannot be defined. 

Assume, you have the relation (Eigenwertgleichung)

Ax = ax

Here, A is an Operator (for instance the Hamilton Operator H) or more generally, a Matrix.

The number a is the Eigenvalue (if A = H , than a = En - the energy values of the Operator).In physics, the Operator corresonds to an physical quantity, a the measurable value. x corresponds to the Eigenvector (in the case of H the wave function), more generally x is an vector.

Once you know the eigenvalues and their corresponding eigenvectors of a matrix it is possible to calculate the result of applying this very matrix to any other vector (after decomposing it into eigenvectors, which form a complete basis).  This may be, of course, calculated directly as well.  In other words, the states (vectors) being proportional to eigenvectors remain stable under the operation of a said matrix.  If you want to call 'stability' an 'equilibrium' - then you have an answer.

Consider a wine glass of standard shape and size. With your index, knock on the side of the glass. The sound you hear are the eigen-frequencies (eigen-values) and the deformations of the glass needed to generate the eigen-frequencies are the eigen-vectors.  

I thought, the following link could be useful:

https://en.wikipedia.org/wiki/Normal_mode

As given in the link, two particles (with equal mass m) are connected by a spring (whose masses are neglected) with spring constant k. In one dimensional motion(say, along x-axis), the system has eigen values w1 and w2 with respective eigen vectors (1,1) and (1,-1). If the system has frequency w1, both particles move in the same directions represented by eigen vector (1,1). If the system has frequency  w2, both particles move in the opposite directions (say, the first mass moves along positive x-axis, the second mass moves along negative x-axis) represented by eigen vector (1,-1).

Thank you for your answer Mr. Mahendra Thapa