Class 6

Beginning of Eigenvectors and Eigenvalues/Chapter 5 - if time is available

• Definition: Ax=cx, x not 0 (x - eigenvector, c - eigenvalue)
• Geometry: See the 3-minute demo on 2 by 2 matrices.
• Definition: given eigenvalue c of A, the eigenspace of A corresponding to c is:
  the null space of (A-cI),
  i.e. it is the set of all eigenvectors with the zero vector added in.
• Examples with n=2 and n=3.
• MATLAB example (you can cut and paste this into MATLAB):

  A=[
  2 0 0
  -1 3 1
  -1 1 3]
  disp('all eigenvalues')
  lambdas=eig(A)
  lambda1= lambdas(1);
  D1=A-lambda1*eye(3)
  rref(D1)
  disp('eigenvector for first eigenvalue')
  v1=[-ans(1:2,3)
        eye(1)]
  disp('check eigenvector equation, i.e. norm of Av1-lambda1 v1')
  norm(A*v1-lambda1*v1)
  disp('now for second eigenvalue')
  lambda2= lambdas(2);
  D2=A-lambda2*eye(3)
  rref(D2)
  disp('two linearly indep. eigenvectors for second eigenvalue')
  v2=[-ans(1,2:3)
        eye(2)]
  disp('check eigenvector equation, i.e. norm of Av2-lambda1 v2')
  norm(A*v2-lambda2*v2)