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258 Chapter Eight: Eigenvectors and Eigenvalues
In order to clarify the definition and illustrate the tech-
nique for finding eigenvectors and eigenvalues, an example will
be worked out in detail.
Example 8.1.1
Consider the real 2 x 2 matrix
A - 1
<1 4
The condition for the vector
'xx^
M 2)
x
to be an eigenvector of A is that AX = cX for some scalar
c. This is equivalent to (A — cI 2)X = 0, which simply asserts
that X is a solution of the linear system
2 - c - 1 Xi
2 4 - c %2
Now by 3.3.2 this linear system will have a non-trivial solution
xi, X2 if and only if the determinant of the coefficient matrix
vanishes,
2 - c - 1
= 0,
2 4 - <
2
that is, c — 6c + 10 = 0. The roots of this quadratic equa-
tion are c\ = 3 + >/^T and C2 = 3 — -\f—l, so these are the
eigenvalues of A.
The eigenvectors for each eigenvalue are found by solving
the linear systems (A — C\I 2)X = 0 and (A — c 2l2)X = 0. For
example, in the case of c\ we have to solve
{-\-4^l)x X- £2=0
=
2zi + (l->/ l)a;2 = 0