5.1 Eigenvectors & Eigenvalues
                                 Math 2331 – Linear Algebra
                                     5.1 Eigenvectors & Eigenvalues
                                                          Jiwen He
                                  Department of Mathematics, University of Houston
                                                  jiwenhe@math.uh.edu
                                         math.uh.edu/∼jiwenhe/math2331
        Jiwen He, University of Houston                     Math 2331, Linear Algebra                                       1 / 14
                               5.1 Eigenvectors & Eigenvalues      Definition Eigenspace Matrix Powers Triangular Matrix
                               5.1 Eigenvectors & Eigenvalues
                  Eigenvectors & Eigenvalues
                  Eigenspace
                  Eigensvalues of Matrix Powers
                  Eigensvalues of Triangular Matrix
                  Eigenvectors and Linear Independence
        Jiwen He, University of Houston                   Math 2331, Linear Algebra                                      2 / 14
                               5.1 Eigenvectors & Eigenvalues      Definition Eigenspace Matrix Powers Triangular Matrix
     Eigenvectors & Eigenvalues: Example
          The basic concepts presented here - eigenvectors and eigenvalues -
          are useful throughout pure and applied mathematics. Eigenvalues
          are also used to study difference equations and continuous
          dynamical systems. They provide critical information in
          engineering design, and they arise naturally in such fields as
          physics and chemistry.
          Example
                              0 −2                    1                       −1 
          Let A =           −4          2 , u =            1 , and v =                  1 . Examine the
          images of u and v under multiplication by A.
          Solution                                                                       
                   Au=              0 −2              1      = −2 =−2 1 =−2u
                                 −4         2         1              −2                     1
          u is called an eigenvector of A since Au is a multiple of u.
        Jiwen He, University of Houston                   Math 2331, Linear Algebra                                      3 / 14
                               5.1 Eigenvectors & Eigenvalues      Definition Eigenspace Matrix Powers Triangular Matrix
     Eigenvectors & Eigenvalues: Example (cont.)
                             Av =            0 −2  −1 = −2 6=λv
                                           −4         2            1               6
          v is not an eigenvector of A since Av is not a multiple of v.
                                             Au=−2u,but Av6=λv
        Jiwen He, University of Houston                   Math 2331, Linear Algebra                                      4 / 14