MATLAB Function Reference Volume 1: A - E - Bad Request

alance
Examples This example shows the basic idea. The matrix A has large elements in the
upper right and small elements in the lower left. It is far from being symmetric.
2-124
A = [1 100 10000; .01 1 100; .0001 .01 1]
A =
1.0e+04 *
0.0001 0.0100 1.0000
0.0000 0.0001 0.0100
0.0000 0.0000 0.0001
Balancing produces a diagonal T matrix with elements that are powers of two
and a balanced matrix B that is closer to symmetric than A.
[T,B] = balance(A)
T =
1.0e+03 *
2.0480 0 0
0 0.0320 0
0 0 0.0003
B =
1.0000 1.5625 1.2207
0.6400 1.0000 0.7813
0.8192 1.2800 1.0000
To see the effect on eigenvectors, first compute the eigenvectors of A.
[V,E] = eig(A); V
V =
-1.0000 0.9999 0.9937
0.0050 0.0100 -0.1120
0.0000 0.0001 0.0010
Note that all three vectors have the first component the largest. This indicates
V is badly conditioned; in fact cond(V) is 8.7766e+003. Next, look at the
eigenvectors of B.
[V,E] = eig(B); V
V =
-0.8873 0.6933 0.0898
0.2839 0.4437 -0.6482
0.3634 0.5679 -0.7561