what is geometrical meaning of a matrix,eigen value?

Answer 1

Okay, a matrix can also be treated as a transformation of a vector, as the equation below illustrates:

v2 = A v1

where A is the matrix and v1 and v2 are before and after vectors. If the vector v2 is in the same direction as vector v1 after such transformation, it's called the eigenvector, and the ratio of lengths of before and after such vectors is called the eigenvalue. So, if the eigenvalue of an eigenvector of a matrix is 1, for instance, then that eigenvector is completely unchanged by the application of that matrix.

Eigenvalue studies are particularly important in determining quantum electron orbitals in atoms, for example, because orbital functions are stable in time, "unaffected by transformation".

Answer 2

An eigenvector is a special direction that unchanged when transformed by the matrix. The length of the eigenvector is changed by by eigenvalue but the vector points in the same direction.

A matrix that projects all vectors (x,y) to (x,0) has two eigenvalues, and eigenvectors:

P=
1 0
0 0

See how any P:(x,y) ->(x,0) ?

1. for eigenvalue = 1, all vectors (x,0) -> (x,0) , they stay the same, the x axis is unchanged.
2. for eigenvalue = 0, all vectors (0,y)->(0,0), the y axis is mapped to zero.

Btw, if a matrix has zero as an eigenvalue it means that it is not invertible and there's a direction (vector) that is mapped to zero. Of course, for our example, this direction is the y axis.