1. 5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors

2. For eigenvalue eigenvector

3. Definition An eigenvector of an matrix A is a nonzero vector x such that for some scalar. A scalar is called an eigenvalue of A if there is a nontrivial solution x of ; such an x is called an eigenvector corresponding to. Example: Are u and v eigenvectors of A ?

4. Example: Show that 7 is an eigenvalue of and find the corresponding eigenvectors.

5. Definition The set of all solutions of is called the eigenspace of A corresponding to. Note: 1.Eigenspace of A is a null space of. 2. Eigenspace of A is a subspace of for matrix A.

6. Example: Find the basis for the corresponding eigenspace of to eigenvalue 2.

7. Theorem 1 The eigenvalues of a triangular matrix are the entries on its main diagonal.

8. Example: Find the eigenvalues of

9. Theorem 2 If are eigenvectors that correspond to distinct eigenvalues of an matrix A, then the set is linearly independent.