Eigenvectors and eigenvaluesThroughout,A will be an n × n matrix.Definition 1. An eigenvector of A is a no nzero x such thatAx = λx for some scalar λ.The scalar λ is the corresponding eigenvalue.In words:Example 2. Verify that1−2is an eigenvec t or of A =0 −2−4 2.Solution.Example 3. U se your geometric understanding to find the eigenvectors and eigenvaluesof A =0 11 0.Solution.Example 4. U se your geometric understanding to find the eigenvectors and eigenvaluesof A =1 00 0.Solution.Armin [email protected] 5. Let P be projection matrix onto the subsp a c e V . What are the eigenvaluesand eigenvectors ofP ?Solution.How to solv e Ax = λxKey observati on:Ax = λxThis has a nonzero solutionRecipe. To find eigenv e ctors and eigenvalues of A.• First, find the eigenvalues λ using:λ is an eigenvalue of Adet (A − λI) = 0• Then, for each eigenvalue, find the eigenvectors byExample 6. Find the eigenvectors and eigenvalues ofA =3 11 3.Solution.Armin [email protected] 7. Find the eigenvectors and eigenvalues ofA =0 −2−4 2.Solution.Example 8. Find the eigenvectors and the eigenvalues ofA =3 2 30 6 100 0 2.Solution.The eigenvalues of a triangular matrix are its diagonal entries.Armin [email protected] 9. Find the eigenvectors and eigenvalues ofA =2 0 0−1 3 1−1 1 3.Solution.Theorem 10. If x1,, xmare eigenvectors of A correspondin g to d ifferent eigenvalues,then they are independent.Why?Review. Ax = λx• To find the eige nvalues λ of A, we use det (A − λ I) = 0.◦ det (A − λI ) is the characteristic polynomial of A.◦ If A is n × n , then the characteristic polynomial has degree n.• Then, for each eigen value, solve (A − λI)x = 0 to find the eigenvectors.Two sources of tr ouble: eigenv alues can be••Armin [email protected] 11. Find the eige nvectors and eigenvalues of A =0 −11 0. Geometrically,what is the trou ble?Solution.Example 12. Find the eigenve c tors and eigenval ues of A =1 10 1. What is the trouble?Solution.Practice problemsExample 13. Find the eigenvectors and eigenvalues of A =0 1−6 5.Example 14. What are the eigenvalues of A =2 0 0 0−1 3 0 0−1 1 3 00 1 2 4?No calculations!Example 15. Find the eigenvectors and eigenvalues of A =1 2 10 −5 01 8 1.Armin
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