When you throw a ball against ground it will bounce back but need not be in same ( i.e opposite) direction. If a ball hit a horizontal ground with angle @ it will bounce back with angle of 180-@. The amount of regained momentum of the ball depends on the coefficient of restitution(COR) of the collision. Neglect the gravity. So here the COR can be considered the 2D scalar field of the collision problem i.e A=(1, 0 || 0, -COR)

The momentum of the ball is considered as some vector(V) in a xy plan where y axis is normal to the ground. When the vector V is multiplied with the scalar matrix(A) the V is reflected about the y axis in xy plan. The new vector V2 will be differs from the V. When @ becomes 90(V=k*[0,1]), the ball will bounce back in same path but opposite direction with V2= k*COR*[0,-1]. Here the V2 and V are parallel but opposite. V direction not changed by A.So V can be considered as eigenvector of the A. COR is eigenvalue.

(This explanation can be wrong but still i tried to explain with my knowledge)

classical examples: principal stress, natural frequency of the spring mass system

Thank you sathhish... I will try understand and get back to you