Mathematical Methods for Physicists: A concise introduction - Site Map

EIGENVECTORS OF COMMUTING MATRICES
then we would have obtained (verify)
0
5 0
1
0
~S 1 B ~A ~S ˆ @ 0 1
C
0A:
0 0 1
Example 3.17
Show that the matrix
is not diagonalizable.
~A ˆ 3 2
2 1
Solution:
The characteristic equation of ~A is
‡ 3 2
2 1 ˆ… ‡ 1†2 ˆ 0:
Thus ˆ1 the only eigenvalue of ~A; the eigenvectors corresponding to ˆ1
are the solutions of
!
‡ 3 2 x1
ˆ 0
) 2 2 ! x1
ˆ 0
2 1 x 2 0 2 2 x 2 0
from which we have
2x 1 2x 2 ˆ 0;
2x 1 2x 2 ˆ 0:
The solutions to this system are x 1 ˆ t; x 2 ˆ t; hence the eigenvectors are of the
form
t 1
ˆ t :
t 1
A does not have two linearly independent eigenvectors, and is therefore not
diagonalizable.
Eigenvectors of commuting matrices
There is a theorem on eigenvectors of commuting matrices that is of great importance
in matrix algebra as well as in quantum mechanics. This theorem states that:
Two commuting matrices possess a common set of eigenvectors.
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