Mathematical Methods for Physicists: A concise introduction - Site Map

THE MATRIX EIGENVALUE PROBLEM
is an eigenvector of A ~ corresponding to 1 ˆ 6. In the same way we ®nd the
eigenvector corresponding to 2 ˆ 1:
1
X 2 ˆ :
1
Example 3.13
If ~A is a non-singular matrix, show that the eigenvalues of ~A 1 are the reciprocals
of those of ~A and every eigenvector of ~A is also an eigenvector of ~A 1 .
Solution:
that
Let be an eigenvalue of ~A corresponding to the eigenvector X, so
~AX ˆ X:
Since ~A 1 exists, multiply the above equation from the left by ~A 1
~A 1 ~AX ˆ ~A 1 X ) X ˆ ~A 1 X:
Since ~A is non-singular, must be non-zero. Now dividing the above equation by
, we have
~A 1 X ˆ…1=†X:
Since this is true for every value of ~A, the results follows.
Example 3.14
Show that all the eigenvalues of a unitary matrix have unit magnitude.
Solution: Let ~U be a unitary matrix and X an eigenvector of ~U with the eigenvalue
, so that
~UX ˆ X:
Taking the hermitian conjugate of both sides, we have
X y ~U y ˆ *X y :
Multiplying the ®rst equation from the left by the second equation, we obtain
X y ~U y ~UX ˆ *X y X:
Since ~U is unitary, ~U y ~U= ~I, so that the last equation reduces to
X y X…jj 2 1† ˆ0:
Now X y X is the square of the norm of X and hence cannot vanish unless X is a
null vector and so we must have jj 2 ˆ 1orjj ˆ1; proving the desired result.
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