Database Reference
In-Depth Information
matrix is symmetric if the element in row
i
and column
j
equals the element in row
j
and
column
i
.
11.1.1
Definitions
Let
M
be a square matrix. Let λ be a constant and
e
a nonzero column vector with the same
number of rows as
M
. Then λ is an
eigenvalue
of
M
and
e
is the corresponding
eigenvector
of
M
if
M
e
= λ
e
.
If
e
is an eigenvector of
M
and
c
is any constant, then it is also true that
c
e
is an ei-
genvector of
M
with the same eigenvalue. Multiplying a vector by a constant changes the
length of a vector, but not its direction. Thus, to avoid ambiguity regarding the length, we
shall require that every eigenvector be a
unit vector
, meaning that the sum of the squares
of the components of the vector is 1. Even that is not quite enough to make the eigenvector
unique, since we may still multiply by −1 without changing the sum of squares of the com-
ponents. Thus, we shall normally require that the first nonzero component of an eigenvec-
tor be positive.
EXAMPLE
11.1 Let
M
be the matrix
One of the eigenvectors of
M
is
and its corresponding eigenvalue is 7. The equation
demonstrates the truth of this claim. Note that both sides are equal to
Also observe that the eigenvector is a unit vector, because
1/5 + 4/5 = 1.
□
11.1.2
Computing Eigenvalues and Eigenvectors
We have already seen one approach to finding an
eigenpair
(an eigenvalue and its corres-
ponding eigenvector) for a suitable matrix
M
in
Section 5.1
: start with any unit vector
v
