Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Vector Calculus
Lecture Slides Set 1
Examples
Application
Eigenspace
Determining the eigenspace
Example
Introduction Eigenvalues and Eigenvectors
Determining the
eigenvalues
Example
Remy Spliet
Erasmus University Rotterdam
Vectors and Matrix multiplication
Consider the matrix A =
5
such that Ax =
.
7
1
3
3
1
and the vector x =
Vector Calculus
2
1
,
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Vectors and Matrix multiplication
1 3
Consider the same matrix A =
but this time the vector
3 1
1
4
x=
, such that Ax =
. Observe: Ax = 4x.
1
4
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Eigenvalues and
eigenvectors
Example
Determining the
eigenvalues
Example
Vector Calculus
Eigenvalues and eigenvectors
Introduction
Eigenvalues and
eigenvectors
Definition
Let A be an n × n matrix. A scalar λ is called an eigenvalue of A
if there exists a vector x, such that x ̸= 0 and Ax = λx. Such a
vector x is called an eigenvector of A corresponding to λ.
Examples
Application
Eigenspace
Determining the eigenspace
Example
1 3
Example: For the matrix A =
it holds that 4 is an
3 1
1
is an eigenvector, because
eigenvalue and x =
1
4
Ax =
= 4x.
4
Remark:
An m × n matrix A does not have eigenvectors, since for any
vector x ∈ Rn it holds that Ax = y where y ∈ Rm . Hence, it is
impossible that Ax = λx.
Determining the
eigenvalues
Example
Vector Calculus
Example 1
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Is the below vector x an eigenvector of the below matrix A?
Eigenspace
Determining the eigenspace
Example
A=
0
2
1
1
1
1
1
2
x=
1
2
2
4
Example
Compute:
Ax =
0
2
=
Determining the
eigenvalues
= 2x
Hence, x is an eigenvector of A corresponding to the eigenvalue 2.
Vector Calculus
Example 2
Introduction
Eigenvalues and
eigenvectors
Examples
Is the below vector x an eigenvector of the below matrix A?
Application
Eigenspace
A=
0
2
1
1
x=
1
0
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Compute:
0
Ax =
=
2
0
Because for every λ it holds that λx ̸=
is x not an
2
eigenvector.
0
2
1
1
1
0
Vector Calculus
Example 3
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Is the below vector x an eigenvector of the below matrix A?
Eigenspace
Determining the eigenspace
Example
A=
0
0
0
2
1
1
x=
0
0
is by definition not an eigenvector.
0
(Despite the fact that Ax =
= x.)
0
The vector
Determining the
eigenvalues
Example
Vector Calculus
Example 4
Introduction
Is 3 an eigenvalue of the below matrix A?
5 1
A=
2 4
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
It follows that 3 is an eigenvalue of A if there exists a vector x
such that Ax = 3x.
⇒
Ax
= 3x
Ax
= 3Ix
⇒ (A − 3I)x
=0
Therefore, 3 is an eigenvalue if the null space of A − 3I contains at
least one vector x such that x ̸= 0.
Determining the
eigenvalues
Example
Vector Calculus
Example 4
Compute
Introduction
A − 3I =
5
2
1
4
1
1
−3
1
0
0
1
Eigenvalues and
eigenvectors
Examples
Application
=
2
2
Find the null space by reducing the augmented matrix to row
echelon form:
2 1 0 R2 −R1 2 1 0
→
2 1 0
0 0 0
So, for an arbitrary value t ̸= 0 of x2 , the null space contains the
vector
1 −2
x=t
1
−1
Example: for t = 2 we find the eigenvector
corresponding
2
to the eigenvalue 3.
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Application
Determining the
eigenvalues
Example
Search engine
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
▶ Rank websites according to relevance.
▶ More linked to → more relevant.
▶ Websites linked to by relevant websites are themselves
relevant.
Determining the
eigenvalues
Example
Vector Calculus
Search engine
Introduction
Example:
Linking matrix A: aij = 1 if website i is linked to by website j.
Eigenvalues and
eigenvectors
Examples

0
 1

A=
 0
 1
1
0
0
1
1
1
1
0
0
1
0
0
0
0
0
1
Application

0
0 

1 

0 
0
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
For r, where ri is the relevance of website i, and scaling
parameter α:
r1 = αr3
r2 = αr1
r3 = α(r2 + r5 )
r4 = α(r1 + r2 + r3 )
r5 = α(r1 + r2 + r4 )
(1)
Vector Calculus
Search engine
Introduction
The system (1) can be written as:
Eigenvalues and
eigenvectors
Examples
r
⇒ Ar
= αAr
1
= r
α
Hence, α1 is an eigenvalue of A and the relevance r is the
corresponding eigenvector.
In this example it holds that:


0.14
 0.08 



r=
 0.22 
 0.27 
0.29
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Eigenspace
Determining the
eigenvalues
Example
Eigenspace
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Definition
Let A be an n × n matrix and let λ be an eigenvalue of A. The
collection of all eigenvectors corresponding to λ, together with the
zero vector, is called the eigenspace of λ in A, and is denoted
by Eλ .
Determining the
eigenvalues
Example
Eigenspace - What does the eigenspace look like?
Let A be a 2 × 2 matrix and λ a corresponding eigenvalue.
Moreover, x is an eigenvector corresponding to the eigenvalue λ.
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Eigenspace - What does the eigenspace look like?
It holds for an arbitrary scalar c that also cx is an eigenvector.
Indeed: A(cx) = c(Ax) = cλx = λ(cx).
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Eigenspace - What does the eigenspace look like?
Suppose y is also an eigenvector corresponding to the eigenvalue λ.
(So both Ax = λx and Ay = λy.)
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Eigenspace - What does the eigenspace look like?
It follows that any linear combination of x and y, z = c1 x + c2 y is
also an eigenvector corresponding to the eigenvalue λ. Indeed:
Az = A(c1 x + c2 y) = c1 (Ax) + c2 (Ay) = c1 λx + c2 λy = λz.
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Eigenspace - What does the eigenspace look like?
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
For matrix A with eigenvalue λ such that x1 , . . . , xk are all
eigenvectors, it holds that any linear combination of the vectors
x1 , . . . , xk are also eigenvectors. So:
span(x1 , . . . , xk ) ⊆ Eλ
Example
Determining the
eigenvalues
Example
Determining the eigenspace for a given
eigenvalue
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Let A be a matrix with corresponding eigenvalue λ. The
eigenspace Eλ contains all vectors x for which:
Application
Eigenspace
Determining the eigenspace
Example
Determining the
eigenvalues
⇒
Ax
= λx
Ax
= λIx
⇒ (A − λI)x
=0
Therefore:
The eigenspace Eλ is the null space of (A − λI).
Example
Vector Calculus
Example
Find the eigenspace E5 corresponding to the eigenvalue 5 of
the below matrix A.
Introduction
Eigenvalues and
eigenvectors
Examples
A=
4
1
3
2
Application
Eigenspace
Determining the eigenspace
Example
The eigenspace is the null space of A − 5I. Compute
4 3
1 0
A − 5I =
−5
1 2
0 1
−1 3
=
1 −3
Find the null space by reducing the augmented matrix to row
echelon form:
−1 3 0 R2 +R1 −1 3 0
→
1 −3 0
0 0 0
Determining the
eigenvalues
Example
Example
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Hence, for an arbitrary value t ̸= 0 of x2 we find the (eigen)vector
in the null space
3
x=t
1
3
We conclude E5 = span
.
1
Determining the eigenspace
Example
Determining the
eigenvalues
Example
Vector Calculus
Introduction
Eigenvalues and
eigenvectors
Examples
Application
Eigenspace
Determining the eigenspace
Determining the
eigenvalues
Example
Determining the
eigenvalues
Example
Vector Calculus
Determining the eigenvalues
Let A be an n × n matrix. The value λ is an eigenvalue if...
Introduction
Eigenvalues and
eigenvectors
... ∃x ̸= 0 such that
Examples
Application
Eigenspace
Ax
⇒
Ax
⇒ (A − λI)x
= λx
Determining the eigenspace
Example
= λIx
Determining the
eigenvalues
=0
Example
⇒ ... the null space of A − λI is not empty.
⇒ ... A − λI is not invertible
(According to the fundamental theorem of invertibility (e.g.,
version 2 of Theorem 3.27 in the book.))
⇒ ... the determinant of A − λI is equal to 0
(By Theorem 4.6).
Vector Calculus
Example
Determine all eigenvalues of the below matrix A.
Introduction
Eigenvalues and
eigenvectors
A=
4
1
3
2
Examples
Application
Eigenspace
Determining the eigenspace
Example
We look for values λ such that the determinant of A − λI equals 0.
Determining the
eigenvalues
Example
⇒
⇒
det(A − λI)
4−λ
3
1
2−λ
=0
(4 − λ)(2 − λ) − 3 · 1
=0
2
⇒ λ − (4 + 2)λ + 4 · 2 − 3 · 1
⇒
⇒
λ2 − 6λ + 5
(λ − 1)(λ − 5)
We find two eigenvalues, they are 1 and 5.
=0
=0
=0
=0