Page 346 - 35Linear Algebra
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346                                                                          Sample Final Exam


                              11. A team of distinguished, post-doctoral engineers analyzes the design for a
                                  bridge across the English channel. They notice that the force on the center
                                                                                      
                                                                                       x
                                  of the bridge when it is displaced by an amount X =     is given by
                                                                                       y
                                                                                       z

                                                                             
                                                                    −x − y
                                                          F =    −x − 2y − z    .
                                                                    −y − z

                                  Moreover, having read Newton’s Principiæ, they know that force is propor-
                                  tional to acceleration so that 2
                                                                      2
                                                                     d X
                                                                F =       .
                                                                      dt 2
                                  Since the engineers are worried the bridge might start swaying in the heavy
                                  channel winds, they search for an oscillatory solution to this equation of the
                                  form 3
                                                                          
                                                                           a
                                                            X = cos(ωt)     .
                                                                           b
                                                                           c
                                   (a) By plugging their proposed solution in the above equations the engi-
                                       neers find an eigenvalue problem

                                                                           
                                                                  a           a
                                                                               b
                                                                  b
                                                             M      = −ω 2     .
                                                                  c            c
                                       Here M is a 3 × 3 matrix. Which 3 × 3 matrix M did the engineers
                                       find? Justify your answer.

                                   (b) Find the eigenvalues and eigenvectors of the matrix M.
                                   (c) The number |ω| is often called a characteristic frequency. What char-
                                       acteristic frequencies do you find for the proposed bridge?

                                   (d) Find an orthogonal matrix P such that MP = PD where D is a
                                       diagonal matrix. Be sure to also state your result for D.

                              2
                               The bridge is intended for French and English military vehicles, so the exact units,
                            coordinate system and constant of proportionality are state secrets.
                              3
                               Here, a, b, c and ω are constants which we aim to calculate.

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