Eigenvalues
If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then
the vector is called an eigenvector of that matrix. A vector which is "flipped" to point in the
opposite direction is also considered an eigenvector. Each eigenvector is, in effect, multiplied by
a scalar, called the eigenvalue corresponding to that eigenvector. The eigenspace corresponding
to one eigenvalue of a given matrix is the set of all eigenvectors of the matrix with that
eigenvalue.
Many kinds of mathematical objects can be treated as vectors:
ordered pairs
,
functions
,
harmonic
modes
,
quantum states
, and
frequencies
are examples. In these cases, the concept of
direction
loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract
direction
is unchanged by a given linear transformation, the prefix "eigen" is used, as in
eigenfunction
,
eigenmode
,
eigenstate
, and
eigenfrequency
.
If a
matrix
is a
diagonal matrix
, then its eigenvalues are the numbers on the diagonal and its
eigenvectors are
basis vectors
to which those numbers refer. For example, the matrix
stretches every vector to three times its original length in the x-direction and shrinks every vector
to half its original length in the y-direction. Eigenvectors corresponding to the eigenvalue 3 are
any multiple of the basis vector [1, 0]; together they constitute the eigenspace corresponding to
the eigenvalue 3. Eigenvectors corresponding to the eigenvalue 0.5 are any multiple of the basis
vector [0, 1]; together they constitute the eigenspace corresponding to the eigenvalue 0.5. In
contrast, any other vector, [2, 8] for example, will change direction. The angle [2, 8] makes with
the x-axis has tangent 4, but after being transformed, [2, 8] is changed to [6, 4], and the angle
that vector makes with the x-axis has tangent 2/3.
Linear transformations
of a
vector space
, such as
rotation
,
reflection
, stretching, compression,
shear
or any combination of these, may be visualized by the effect they produce on
vectors
. In
other words, they are vector functions. More formally, in a vector space
L
, a vector function
A
is
defined if for each vector
x
of
L
there corresponds a unique vector
y
=
A
(
x
) of
L
. For the sake of
brevity, the parentheses around the vector on which the transformation is acting are often
omitted. A vector function
A
is
linear
if it has the following two properties:
•
Additivity
:
A
(
x
+
y
) =
A
x
+
A
y
•
Homogeneity
:
A
(α
x
) = α
A
x
where
x
and
y
are any two vectors of the vector space
L
and α is any
scalar
.
[12]
Such a function is
variously called a
linear transformation
,
linear operator
, or
linear
endomorphism
on the space
L
.