Lecture 14
Eigenvalues and Eigenvectors
Suppose that
A
is a square (
n
×
n
) matrix. We say that a nonzero vector
v
is an eigenvector (
ev
)
and a number
λ
is its eigenvalue (
ew
) if
A
v
=
λ
v
.
(14.1)
Geometrically this means that
A
v
is in the same direction as
v
, since multiplying a vector by a
number changes its length, but not its direction.
Matlab
has a builtin routine for finding eigenvalues and eigenvectors:
> A = pascal(4)
> [v e] = eig(A)
The results are a matrix
v
that contains eigenvectors as columns and a diagonal matrix
e
that
contains eigenvalues on the diagonal. We can check this by:
> v1 = v(:,1)
> A*v1
> e(1,1)*v1
Finding Eigenvalues for
2
×
2
and
3
×
3
If
A
is 2
×
2 or 3
×
3 then we can find its eigenvalues and eigenvectors by hand. Notice that Equation
(14.1) can be rewritten as:
A
v

λ
v
=
0
.
(14.2)
It would be nice to factor out the
v
from the righthand side of this equation, but we can’t because
A
is a matrix and
λ
is a number. However, since
I
v
=
v
, we can do the following:
A
v

λ
v
=
A
v

λI
v
= (
A

λI
)
v
=
0
(14.3)
If
v
is nonzero, then by Theorem 3 in Lecture 10 the matrix (
A

λI
) must be singular. By the
same theorem, we must have:
det(
A

λI
) = 0
.
(14.4)
This is called the
characteristic equation
.
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 Fall '08
 Young,T
 Eigenvectors, Vectors, Matrices, Complex number, Diagonal matrix, Orthogonal matrix

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