Page 277 - 35Linear Algebra
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                                         Diagonalizing Symmetric Matrices






                   Symmetric matrices have many applications. For example, if we consider the
                   shortest distance between pairs of important cities, we might get a table like
                   the following.


                                                  Davis Seattle San Francisco
                                      Davis          0     2000          80
                                      Seattle      2000      0          2010
                                  San Francisco     80     2010          0



                   Encoded as a matrix, we obtain


                                                                 
                                                 0    2000    80
                                                                         T
                                        M =    2000    0    2010   = M .
                                                 80   2010     0



                                                                 T
                   Definition A matrix M is symmetric if M = M.



                      One very nice property of symmetric matrices is that they always have
                   real eigenvalues. Review exercise 1 guides you through the general proof, but
                   below is an example for 2 × 2 matrices.


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