Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
Once the eigenvalues have been found, corresponding eigenvectors can be
found from the system (3.55). Since the system is homogeneous, if X is an
eigenvector of ~A, then kX, where k is any constant (not zero), is also an eigenvector
of ~A corresponding to the same eigenvalue. It is very easy to show this.
Since ~AX ˆ X, multiplying by an arbitrary constant k will give k ~AX ˆ kX.
Now k ~A ˆ ~Ak (every matrix commutes with a scalar), so we have
~A…kX† ˆ…kX†; showing that kX is also an eigenvector of ~A with the same
eigenvalue . But kX is linearly dependent on X, and if we were to count all
such eigenvectors separately, we would have an in®nite number of them. Such
eigenvectors are therefore not counted separately.
A matrix of order n does not necessarily have n linearly independent
eigenvectors; some of them may be repeated. (This will happen when the characteristic
polynomial has two or more identical roots.) If an eigenvalue occurs m
times, m is called the multiplicity of the eigenvalue. The matrix has at most m
linearly independent eigenvectors all corresponding to the same eigenvalue. Such
linearly independent eigenvectors having the same eigenvalue are said to be degenerate
eigenvectors; in this case, m-fold degenerate. We will deal only with those
matrices that have n linearly independent eigenvectors and they are diagonalizable
matrices.
Example 3.12
Find (a) the eigenvalues and (b) the eigenvectors of the matrix
~A ˆ 5 4
:
1 2
Solution:
(a) The eigenvalues: The characteristic equation is
det… ~A ~I† ˆ 5 4
1 2 ˆ 2 7 ‡ 6 ˆ 0
which has two roots
1 ˆ 6 and 2 ˆ 1:
(b) The eigenvectors: For ˆ 1 the system (3.55) assumes the form
x 1 ‡ 4x 2 ˆ 0;
x 1 4x 2 ˆ 0:
Thus x 1 ˆ 4x 2 , and
X 1 ˆ 4
1
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