Eigen Values and Eigen Vectors

Eigen Values and Eigen Vectors:

For a square matrix ( A ), an eigenvector ( v ) is a nonzero vector that remains in the same direction after multiplication by ( A ):

[ A v = lambda v ]

where:

• ( lambda ) is the eigenvalue, a scalar that represents how much the eigenvector is stretched or shrunk.

• ( v ) is the eigenvector, which defines a direction that remains unchanged under the transformation.

How to Compute Eigenvalues and Eigenvectors

1. Find Eigenvalues:

   • Solve the characteristic equation:
     [ det(A - lambda I ) = 0 ]

   • The solutions ( \lambda ) are the eigenvalues.

2. Find Eigenvectors:

• For each eigenvalue ( lambda ), solve: [ (A - lambda I) v = 0 ]

• The nonzero solutions ( v ) are the eigenvectors.