Page 277 - 35Linear Algebra
P. 277
15
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.
277
