Outline
Week 10: Eigenvalues and eigenvectors
Course Notes: 6.1
Goals: Understand how to find eigenvector/eigenvalue pairs, and use them
to simplify calculations involving matrix powers.
When Matrix Multiplication looks like Scalar Multiplication
2 0
0 2
x
x
=2
y
y
When Matrix Multiplication looks like Scalar Multiplication
2 0
0 2
x
x
=2
y
y
Recall: two vectors that are scalar multiples of one another are parallel.
Reflections, Revisited
y =x
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
0 1
1 0
x
x
=1·
x
x
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
0 1
1 0
x
x
=1·
x
x
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
0 1
1 0
x
x
=1·
x
x
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
0 1
1 0
x
x
=1·
x
x
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
0 1
1 0
x
x
=1·
x
x
Reflections, Revisited
y =x
π
4
cos(2 π4 ) sin(2 π4 )
0 1
Ref π4 =
=
sin(2 π4 ) − cos(2 π4 )
1 0
0 1
1 0
x
x
=1·
x
x
x
0 1
x
= −1 ·
1 0 −x
−x
Eigenvectors and Eigenvalues
0 1 x
x
=1·
1 0 x
x
0 1
1
The matrix
has eigenvector
with eigenvalue 1.
1 0
1
0 1
x
x
= −1 ·
1 0 −x
−x
0 1
−1
Tthe matrix
has eigenvector
with eigenvalue −1.
1 0
1
Given a matrix A, a scalar λ, and a NONZERO vector x with
Ax = λx
we say x is an eigenvector of A with eigenvalue λ.
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
θ
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
θ
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
θ
Rotation Matrix Eigenvalues
cos θ − sin θ
Rotθ =
sin θ cos θ
Search for eigenvectors:
cos θ − sin θ x
x
=λ
sin θ cos θ
y
y
θ
In general, no (real) eigenvalues of Rotθ .
Finding Eigenvectors, Given Eigenvalues


1 2 4
Given: 7 is an eigenvalue of the matrix 2 4 1.
4 1 2
What is an eigenvector associated to that eigenvalue?
Finding Eigenvectors, Given Eigenvalues


1 2 4
Given: 7 is an eigenvalue of the matrix 2 4 1.
4 1 2
What is an eigenvector
associated
to
that
eigenvalue?
 
x
An eigenvector y  with eigenvalue 7 would satisfy this equation:
z

 
 
1 2 4
x
x
2 4 1 y  = 7 y 
4 1 2
z
z
Finding Eigenvectors, Given Eigenvalues


1 2 4
Given: 7 is an eigenvalue of the matrix 2 4 1.
4 1 2
What is an eigenvector
associated
to
that
eigenvalue?
 
x
An eigenvector y  with eigenvalue 7 would satisfy this equation:
z

 
 
1 2 4
x
x
2 4 1 y  = 7 y 
4 1 2
z
z
In equation form:
x
2x
4x
+ 2y
+ 4y
+ y
+ 4x
+ z
+ 2z
= 7x
= 7y
= 7z
Finding Eigenvectors, Given Eigenvalues
In equation form:
x
2x
4x
+ 2y
+ 4y
+ y
+ 4x
+ z
+ 2z
= 7x
= 7y
= 7z
Finding Eigenvectors, Given Eigenvalues
In equation form:
x
2x
4x
+ 2y
+ 4y
+ y
+ 4x
+ z
+ 2z
= 7x
= 7y
= 7z
In better equation form:
−6x
2x
4x
+ 2y
− 3y
+ y
+ 4x
+ z
− 5z
= 0
= 0
= 0
Finding Eigenvectors, Given Eigenvalues
In equation form:
x
2x
4x
+ 2y
+ 4y
+ y
+ 4x
+ z
+ 2z
= 7x
= 7y
= 7z
In better equation form:
−6x
2x
4x
+ 2y
− 3y
+ y
+ 4x
+ z
− 5z
Gaussian elimination on augmented matrix:


−6 2
4
0
 2 −3 1
0 → → →
4
1 −5
0
= 0
= 0
= 0

1 0 −1
0 1 −1
0 0 0

0
0
0
Finding Eigenvectors, Given Eigenvalues
Gaussian elimination on augmented matrix:


−6 2
4
0
 2 −3 1
0 → → →
4
1 −5
0

1 0 −1
0 1 −1
0 0 0

0
0
0
Finding Eigenvectors, Given Eigenvalues
Gaussian elimination on augmented matrix:


−6 2
4
0
 2 −3 1
0 → → →
4
1 −5
0
Solutions:
 
 
x
1
y  = s 1
z
1

1 0 −1
0 1 −1
0 0 0

0
0
0
Finding Eigenvectors, Given Eigenvalues
Gaussian elimination on augmented matrix:


−6 2
4
0
 2 −3 1
0 → → →
4
1 −5
0
Solutions:

1 0 −1
0 1 −1
0 0 0
 
 
x
1
y  = s 1
z
1
So these are the solutions to:

 
 
1 2 4
x
x
2 4 1 y  = 7 y 
4 1 2
z
z

0
0
0
Finding Eigenvectors, Given Eigenvalues
The solutions to:

 
 
1 2 4
x
x
2 4 1 y  = 7 y 
4 1 2
z
z
are precisely the vectors:
 
 
x
1
y  = s 1
z
1
Finding Eigenvectors, Given Eigenvalues
The solutions to:

 
 
1 2 4
x
x
2 4 1 y  = 7 y 
4 1 2
z
z
are precisely the vectors:
 
 
x
1
y  = s 1
z
1
 
1

So, we can choose 1 as an example of an eigenvector with eigenvalue 7.
1
Generalized Eigenvector Finding
Suppose a matrix A has eigenvalue λ. Find an associated eigenvector x .
Generalized Eigenvector Finding
Suppose a matrix A has eigenvalue λ. Find an associated eigenvector x .
Ax = λx
Ax − λx = 0
(A − λI )x = 0
Finding Eigenvectors from Eigenvalues
3 6
The matrix
has eigenvalues −6 and 7.
6 −2
Find an eigenvector associated to each eigenvalue.
Finding Eigenvectors from Eigenvalues
3 6
The matrix
has eigenvalues −6 and 7.
6 −2
Find an eigenvector associated to each eigenvalue.
2
λ = −6, x =
−3
3
λ = 7, x =
2
Finding Eigenvectors from Eigenvalues
3 6
The matrix
has eigenvalues −6 and 7.
6 −2
Find an eigenvector associated to each eigenvalue.
2
λ = −6, x =
−3
Basis of R2 :
3
λ = 7, x =
2
2
3
,
−3
2
Finding Eigenvectors from Eigenvalues
2 3
The matrix
has eigenvalues −2 and 5.
4 1
Find an eigenvector associated to each eigenvalue.
Finding Eigenvectors from Eigenvalues
2 3
The matrix
has eigenvalues −2 and 5.
4 1
Find an eigenvector associated to each eigenvalue.
3
λ = −2, x =
−4
1
λ = 5, x =
1
Finding Eigenvectors from Eigenvalues
2 3
The matrix
has eigenvalues −2 and 5.
4 1
Find an eigenvector associated to each eigenvalue.
3
λ = −2, x =
−4
Basis of R2 :
1
λ = 5, x =
1
3
1
,
−4
1
How do we Find Eigenvalues, Though?
First, a reminder....
Solutions to Systems of Equations
Let A be an n-by-n matrix. The following statements are equivalent:
1) Ax = b has exactly one solution for any b .
2) Ax = 0 has no nonzero solutions.
3) The rank of A is n.
4) The reduced form of A has no zeroes along the main diagonal.
5) A is invertible
6) det(A) 6= 0
invertible
statements 1-4
nonzero determinant
How do we Find Eigenvalues, Though?
A matrix; λ eigenvalue, x eigenvector (so x 6= 0)
How do we Find Eigenvalues, Though?
A matrix; λ eigenvalue, x eigenvector (so x 6= 0)
Ax = λx
Ax − λx = 0
(A − λI )x = 0
(A − λI )x = 0 has a nonzero solution
det(A − λI ) = 0
Find Eigenvalues and Associated Eigenvectors
λ eigenvalue ⇔ det(A − λI ) = 0
1 0
A=
3 −1
1 2
B=
0 1
Find Eigenvalues and Associated Eigenvectors
2
λ = 1,
3
λ eigenvalue ⇔ det(A − λI ) = 0
1 0
A=
3 −1
0
2
0
λ = −1,
,
is a basis of R2
1
3
1
1 2
B=
0 1
Find Eigenvalues and Associated Eigenvectors
2
λ = 1,
3
λ eigenvalue ⇔ det(A − λI ) = 0
1 0
A=
3 −1
0
2
0
λ = −1,
,
is a basis of R2
1
3
1
1 2
B=
0 1
1
λ = 1,
0
NOT a basis of R2 !
Find Eigenvalues and Associated Eigenvectors


1 4 5
A = 0 2 6
0 0 3
Find Eigenvalues and Associated Eigenvectors


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, 0
0
 
4
λ = 2, 1
0
 
29
λ = 3, 12
2
Find Eigenvalues and Associated Eigenvectors


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, 0
0
 
4
λ = 2, 1
0
 
29
λ = 3, 12
2
     
4
29 
 1
0 , 1 , 12 Basis of R3


0
0
2
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1

λ = 1, k1 = 0
0
 
47
x = 16
2
 
4

λ = 2, k2 = 1
0
 
29

λ = 3, k3 = 12
2
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1

λ = 1, k1 = 0
0
 
4

λ = 2, k2 = 1
0
 
29

λ = 3, k3 = 12
2
 
   
 
1
4
29
47
x = 16 = 2 0 + 4 1 + 12 = 2k1 + 4k2 + k3
0
0
2
2
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1

λ = 1, k1 = 0
0
 
4

λ = 2, k2 = 1
0
 
29

λ = 3, k3 = 12
2
 
   
 
1
4
29
47
x = 16 = 2 0 + 4 1 + 12 = 2k1 + 4k2 + k3
0
0
2
2
Ax =
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1

λ = 1, k1 = 0
0
 
4

λ = 2, k2 = 1
0
 
29

λ = 3, k3 = 12
2
 
   
 
1
4
29
47
x = 16 = 2 0 + 4 1 + 12 = 2k1 + 4k2 + k3
0
0
2
2
Ax =A(2k1 + 4k2 + k3 )
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1

λ = 1, k1 = 0
0
 
4

λ = 2, k2 = 1
0
 
29

λ = 3, k3 = 12
2
 
   
 
1
4
29
47
x = 16 = 2 0 + 4 1 + 12 = 2k1 + 4k2 + k3
0
0
2
2
Ax =A(2k1 + 4k2 + k3 )= 2Ak1 + 4Ak2 + Ak3
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1

λ = 1, k1 = 0
0
 
4

λ = 2, k2 = 1
0
 
29

λ = 3, k3 = 12
2
 
   
 
1
4
29
47
x = 16 = 2 0 + 4 1 + 12 = 2k1 + 4k2 + k3
0
0
2
2
Ax =A(2k1 + 4k2 + k3 )= 2Ak1 + 4Ak2 + Ak3 = 2k1 + 4(2)k2 + 3k3
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, k1 = 0
0
 
4
λ = 2, k2 = 1
0
x = 2k1 + 4k2 + k3
A10 x =
 
29
λ = 3, k3 = 12
2
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, k1 = 0
0
 
4
λ = 2, k2 = 1
0
x = 2k1 + 4k2 + k3
A10 x =A10 (2k1 + 4k2 + k3 )
 
29
λ = 3, k3 = 12
2
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, k1 = 0
0
 
4
λ = 2, k2 = 1
0
x = 2k1 + 4k2 + k3
A10 x =A10 (2k1 + 4k2 + k3 )= 2A10 k1 + 4A10 k2 + A10 k3
 
29
λ = 3, k3 = 12
2
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, k1 = 0
0
 
4
λ = 2, k2 = 1
0
 
29
λ = 3, k3 = 12
2
x = 2k1 + 4k2 + k3
A10 x =A10 (2k1 + 4k2 + k3 )= 2A10 k1 + 4A10 k2 + A10 k3 =
2k1 + 4(210 )k2 + 310 k3
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, k1 = 0
0
 
4
λ = 2, k2 = 1
0
 
29
λ = 3, k3 = 12
2
x = 2k1 + 4k2 + k3
10 k + 4A10 k + A10 k =
A10 x =A10 (2k1 + 4k2 + k3 )=
1 
2
3
 2A 14

2
2
29 · 310
2k1 + 4(210 )k2 + 310 k3 = 0 + 212  + 12 · 310 
0
0
2 · 310
Using Eigenvalues to Compute Matrix Powers


1 4 5
A = 0 2 6
0 0 3
 
1
λ = 1, k1 = 0
0
 
4
λ = 2, k2 = 1
0
 
29
λ = 3, k3 = 12
2
x = 2k1 + 4k2 + k3
10 k + 4A10 k + A10 k =
A10 x =A10 (2k1 + 4k2 + k3 )=
1 
2
 2A 14

 3
2
2
29 · 310
2 + 214 + 29 · 310
2k1 + 4(210 )k2 + 310 k3 = 0 + 212  + 12 · 310 =  212 + 12 · 310 
0
0
2 · 310
2 · 310
Random Walks and Eigenvalues
from
to
pass
fail
pass fail
1
2
1
3
1
2
2
3
Random Walks and Eigenvalues
from
to
pass
fail
λ1 = 16 , k1 =
1
;
−1
pass fail
1
2
1
3
1
2
2
3
λ2 = 1, k1 =
2
3
Random Walks and Eigenvalues
from
to
pass
fail
λ1 = 16 , k1 =
1
;
−1
pass fail
1
2
1
3
1
2
2
3
λ2 = 1, k1 =
2
3
1
Suppose your initial state is x0 =
. What happens after n tests?
0
Random Walks and Eigenvalues
from
to
pass
fail
λ1 = 16 , k1 =
1
;
−1
pass fail
1
2
1
3
1
2
2
3
λ2 = 1, k1 =
2
3
1
Suppose your initial state is x0 =
. What happens after n tests?
0
x0 = 53 k1 + 15 k2
Random Walks and Eigenvalues
from
to
pass
fail
λ1 = 16 , k1 =
1
;
−1
pass fail
1
2
1
3
1
2
2
3
λ2 = 1, k1 =
2
3
1
Suppose your initial state is x0 =
. What happens after n tests?
0
x0 = 53 k1 + 15 k2
P n x0 = 35 P n k1 + 15 P n k2 =
3
5
1 n
6
k1 + 15 k2
Random Walks and Eigenvalues
from
to
pass
fail
λ1 = 16 , k1 =
1
;
−1
pass fail
1
2
1
3
1
2
2
3
λ2 = 1, k1 =
2
3
1
Suppose your initial state is x0 =
. What happens after n tests?
0
x0 = 53 k1 + 15 k2
P n x0 = 35 P n k1 + 15 P n k2 =
1
2/5
lim P n x0 = k2 =
n→∞
3/5
5
3
5
1 n
6
k1 + 15 k2
Complex Eigenvalues
0 1
A=
−1 0
Calculate A90 x.
2
x=
3
Complex Eigenvalues
0 1
A=
−1 0
2
x=
3
Calculate A90 x.
−i
λ1 = i, k1 =
1
i
λ2 = −i, k2 =
1
Complex Eigenvalues
0 1
A=
−1 0
2
x=
3
Calculate A90 x.
−i
λ1 = i, k1 =
1
i
λ2 = −i, k2 =
1
x = (1.5 + i)k1 + (1.5 − i)k2
Complex Eigenvalues
0 1
A=
−1 0
2
x=
3
Calculate A90 x.
−i
λ1 = i, k1 =
1
i
λ2 = −i, k2 =
1
x = (1.5 + i)k1 + (1.5 − i)k2
A90 x = (1.5 + i)A90 k1 + (1.5 − i)A90 k2
= (1.5 + i)(i)90 k1 + (1.5 − i)(−i)90 k2
−2
= (1.5 + i)(−1)k1 + (1.5 − i)(−1)k2 =
−3
Complex Eigenvalues: A Neat Trick
0 1
A=
−1 0
−i
λ1 = i, k1 =
1
2
x=
3
i
λ2 = −i, k2 =
1
Complex Eigenvalues: A Neat Trick
0 1
A=
−1 0
−i
λ1 = i, k1 =
1
2
x=
3
i
λ2 = −i, k2 =
1
The eigenvalues and eigenvectors are complex conjugates of one another.
Complex Eigenvalues: A Neat Trick
0 1
A=
−1 0
−i
λ1 = i, k1 =
1
2
x=
3
i
λ2 = −i, k2 =
1
The eigenvalues and eigenvectors are complex conjugates of one another.
Ax = λx
⇔
Ax = λx
⇔
Ax = λx
⇔
Ax = λx
Complex Eigenvalues: A Neat Trick
Find all eigenvalues and eigenvectors of :
−3 5
A=
−2 −1
Complex Eigenvalues: A Neat Trick
Find all eigenvalues and eigenvectors of :
−3 5
A=
−2 −1
−3 5
λ 0
−3 − λ
5
det
−
= det
= λ2 + 4λ + 13
−2 −1
0 λ
−2
−1 − λ
p
−4 ± 16 − 4(13)
Roots:
= −2 ± 3i
2
Complex Eigenvalues: A Neat Trick
Find all eigenvalues and eigenvectors of :
−3 5
A=
−2 −1
−3 5
λ 0
−3 − λ
5
det
−
= det
= λ2 + 4λ + 13
−2 −1
0 λ
−2
−1 − λ
p
−4 ± 16 − 4(13)
Roots:
= −2 ± 3i
2
λ1 = −2 − 3i,
Complex Eigenvalues: A Neat Trick
Find all eigenvalues and eigenvectors of :
−3 5
A=
−2 −1
−3 5
λ 0
−3 − λ
5
det
−
= det
= λ2 + 4λ + 13
−2 −1
0 λ
−2
−1 − λ
p
−4 ± 16 − 4(13)
Roots:
= −2 ± 3i
2
λ1 = −2 − 3i,
x1 =
1 + 3i
2
Complex Eigenvalues: A Neat Trick
Find all eigenvalues and eigenvectors of :
−3 5
A=
−2 −1
−3 5
λ 0
−3 − λ
5
det
−
= det
= λ2 + 4λ + 13
−2 −1
0 λ
−2
−1 − λ
p
−4 ± 16 − 4(13)
Roots:
= −2 ± 3i
2
λ1 = −2 − 3i,
λ2 = −2 + 3i,
x1 =
1 + 3i
2
Complex Eigenvalues: A Neat Trick
Find all eigenvalues and eigenvectors of :
−3 5
A=
−2 −1
−3 5
λ 0
−3 − λ
5
det
−
= det
= λ2 + 4λ + 13
−2 −1
0 λ
−2
−1 − λ
p
−4 ± 16 − 4(13)
Roots:
= −2 ± 3i
2
λ1 = −2 − 3i,
1 + 3i
2
1 − 3i
x2 =
2
x1 =
λ2 = −2 + 3i,