Although the process of applying a linear operator
T to a vector gives a vector in the same space as the original, the resulting vector usually points in a completely different direction from the original, that is,
T(
x) is neither parallel nor antiparallel to
x. However, it can happen that
T(
x)
is a scalar multiple of
x—even when
x ≠ 0—and this phenomenon is so important that it deserves to be explored.
If
T :
R
n
→
R
n
is a linear operator, then
T must be given by
T(
x) =
A
x for some
n x n matrix
A. If
x ≠ 0 and
T(
x) =
A
x is a scalar multiple of
x, that is, if
for some scalar λ, then λ is said to be an
eigenvalue of
T (or, equivalently, of
A). Any
nonzero vector
x which satisfies this equation is said to be an
eigenvector of
T (or of
A) corresponding to λ. To illustrate these definitions, consider the linear operator
T :
R2 →
R2 defined by the equation
That is,
T is given by left multiplication by the matrix
Consider, for example, the image of the vector
x = (1, 3)T under the action of
T:
Clearly,
T(
x) is not a scalar multiple of
x, and this is what typically occurs.
However, now consider the image of the vector
x = (2, 3)T under the action of
T:
Here,
T(
x)
is a scalar multiple of
x, since
T(
x) = (−4, −6)T = −2(2, 3)T = −2
x. Therefore, −2 is an eigenvalue of
T, and (2, 3)T is an eigenvector corresponding to this eigenvalue. The question now is, how do you determine the eigenvalues and associated eigenvectors of a linear operator?
Vector Algebra
Eigenvalues and Eigenvectors
