1
Math 1b Practical — Eigenvalues and eigenvectors, II
February 11, 2011
Example 6.
Let
P
be the matrix of the orthogonal projection onto a subspace
U
of
R
n
.
If
u
∈
U
, then
P
u
=
u
, and if
w
∈
U
⊥
, then
P
w
= 0. That is, elements of
U
are eigenvec
tors corresponding to eigenvalue 1, and elements of
U
⊥
are eigenvectors corresponding to
eigenvalue 0. The (geometric) multiplicity of 1 is dim(
U
) and the (geometric) multiplicity
of 0 is
n
−
dim(
U
).
Let
R
be the matrix of the reﬂection through a subspace
U
of
R
n
. If
u
∈
U
, then
R
u
=
u
, and if
w
∈
U
⊥
, then
R
w
=
−
w
.
That is, elements of
U
are eigenvectors
corresponding to eigenvalue +1, and elements of
U
⊥
are eigenvectors corresponding to
eigenvalue
−
1. (I may give a more careful explanation in class.)
Example 7.
If
A
and
B
are square matrices, the eigenvalues of
A
O
O
B
are those of
A
AND those of
B
.
More about this in class, or in a later version of this
handout.
∗
∗
∗
In Example 1 of the part I of this handout, we saw that 5 and
−
2 were the eigenvalues
of
A
=
1 3
4 2
and that
A
e
1
=
1
3
4
2
3
4
=
15
20
= 5
e
1
,
A
e
2
=
1
3
4
2
1
−
1
=
−
2
2
=
−
2
e
2
.
As an indication of why this helps us understand
A
and its powers, first note that
A
n
e
1
= 5
n
e
1
and
A
n
e
2
= (
−
2)
n
e
2
.
Since
e
1
and
e
2
form a basis for
R
2
, we can understand
A
n
x
by writing
x
as a linear
combination of
e
1
and
e
2
. For example, let
x
= (9
,
5)
. We have (check this)
9
5
= 2
3
4
+ 3
1
−
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
and then
A
n
9
5
= 2
A
n
3
4
+3
A
n
1
−
1
= 2
·
5
n
3
4
+3
·
(
−
2)
n
1
−
1
=
6
·
5
n
+ 3
·
(
−
2)
n
8
·
5
n
−
3
·
(
−
2)
n
.
That is, we have derived a
formula
for
A
n
(9
,
5)
. One can see from this, for example, that
the ratio of its first and second coordinates tends to 3
/
4 as
n
tends to infinity.
Sometimes a matrix has an positive eigenvalue
λ
so that
λ >

μ

for every other
eigenvalue
μ
; such an eigenvalue is called the
dominant eigenvalue
. In this example, 5 is
the dominant eigenvalue.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Aschbacher
 Linear Algebra, Eigenvectors, Vectors, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, FN

Click to edit the document details