Eigenvalues & Eigenvectors

Eigenvalues & Eigenvectors

1. Concepts

   • • Eigenvalues are also called proper values, characteristic values, latent values
     • Eigenvector cannot be 0, but eigenvalue possible to be 0

2. Eigenspaces

   • if A is an n x n matrix with an eigenvalue and corresponding eigenvectors, then every nonzero scalar multiple of eigenvector is also an eigenvector of A
   • Also true that if eigenvectors are corresponding to the same eigenvalues, then their sum is also an eigenvector corresponding to eigenvalue
   • Set of all eigenvectors of a given eigenvalues, zero vectors, is the subspace of which is called the eigenspace of eigenvalue (lambda)

3. Finding Eigenvalues & Eigenvectors

   • • Has nonzero solutions iff the coefficient matrix is not invertible ; det = 0
   • Let A be n x n matrix A
     • An eigenvalue of A is scalar lambda
       • • characteristic equation of A
       • • expanded ; characteristic polynomial of A
     • Eigenvectors of A corresponding to lambda are nonzero solutions of :
   • The determinant
     • • Characteristic polynomial of A
     • • characteristic equation of A
   • Matrix A is singular iff lambda = 0 is an eigenvalue of A
   • The eigenvalues of A =[aij] are the roots of the characteristic polynomial of A
   • The eigenvalues problem in general case
     • • since is nonzero, then it has a nontrivial solution if
   • The null space of is called the eigenspace of A corresponding to the eigenvalue lambda

4. Method to Obtain the Bases of Eigenspace A

   • Step 1 : FInd all eigenvalues of A
   • Step 2 : Find a basis for null space by solving the homogeneous system

5. Finding Eigenvalues of A^k

   • Let lambda be an eigenvalue of A and x bar the corresponding eigenvector
     • • Which means that lambda^3 is an eigenvalue of A^3 and x bar is the corresponding vector
   • A square matrix A is invertible iff lambda = 0 is not an eigenvalue of A
   • Assume that A is a square matrix with eigenvalues

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