MAT 540 - Spring 2012

                   Homework 1 Solutions

                               Ryan Szypowski

                             Due April 17, 2012

1. Let

                                            a b
   
                                    A
   
                                            b c

   be a 2 2 real, symmetric matrix. Show that all eigenvalues of A are real. What values

   of b guarantee that it is positive denite.

   Solution: The eigenvalues of A are the roots of the characteristic polynomial

                      P ( )  det(A   I)
   
                             (a   )(c   )  b2
   
                              2  (a  c)   (b2  ac).

   The roots of this polynomial are

        ...