Eigenvalues,
  Eigenvectors,
  and Diagonal-
     ization
    Math 240
  Eigenvalues
  and
  Eigenvectors
  Diagonalization        Eigenvalues, Eigenvectors, and Diagonalization
                                                      Math 240 — Calculus III
                                                           Summer 2013, Session II
                                                      Wednesday, July 24, 2013
  Eigenvalues,                                                      Agenda
  Eigenvectors,
 and Diagonal-
   ization
   Math 240
 Eigenvalues
 and
 Eigenvectors
 Diagonalization
             1. Eigenvalues and Eigenvectors
             2. Diagonalization
   Eigenvalues,                                                                                         Introduction
  Eigenvectors,
  and Diagonal-
     ization
    Math 240
  Eigenvalues
  and                 Next week, we will apply linear algebra to solving differential
  Eigenvectors
  Diagonalization     equations. One that is particularly easy to solve is
                                                                    y′ = ay.
                      It has the solution y = ceat, where c is any real (or complex)
                      number. Viewed in terms of linear transformations, y = ceat is
                      the solution to the vector equation
                                                                  T(y) = ay,                                               (1)
                      where T : Ck(I) → Ck−1(I) is T(y) = y′. We are going to
                      study equation (1) in a more general context.
  Eigenvalues,                                                                               Definition
  Eigenvectors,
  and Diagonal-
     ization
   Math 240        Definition
 Eigenvalues       Let A be an n×n matrix. Any value of λ for which
 and
 Eigenvectors                                            Av=λv
 Diagonalization
                   has nontrivial solutions v are called eigenvalues of A. The
                   corresponding nonzero vectors v are called eigenvectors of A.
                                          y                                   y
                                              v                                    v
                                                      x                                    x
                                                                                       Av=λv
                             Av = λv        v is an eigenvector of A            v is not an eigenvector of A
                            Figure: A geometrical description of eigenvectors in R2.