The document discusses eigenvectors and eigenvalues. It begins by defining diagonal matrices and provides examples. It then states that the goal is to understand diagonalization using eigenvectors and eigenvalues. Diagonalization involves finding a matrix such that when it transforms another matrix, the result is a diagonal matrix. This requires the eigenvectors of the original matrix to be linearly independent. The document provides examples of calculating eigenvectors and eigenvalues from matrices and shows how this relates to diagonalization. It also gives a brief introduction to linear independence and its implications for diagonalization.

1. Eigenvectors and
Eigenvalues
By Christopher Gratton
cg317@exeter.ac.uk

2. Introduction: Diagonal Matrices
Before beginning this topic, we must first
clarify the definition of a “Diagonal Matrix”.
A Diagonal Matrix is an n by n Matrix whose
non-diagonal entries have all the value zero.

3. Introduction: Diagonal Matrices
In this presentation, all Diagonal Matrices will
be denoted as:
where dnn is the n-th row and the n-th
column of the Diagonal Matrix.

4. Introduction: Diagonal Matrices
For example, the previously given Matrix of:
Can be written in the form:
diag(5, 4, 1, 9)

5. Introduction: Diagonal Matrices
The Effects of a Diagonal Matrix
The Identity Matrix is an example of a
Diagonal Matrix which has the effect of
maintaining the properties of a Vector within
a given System.
For example:

6. Introduction: Diagonal Matrices
The Effects of a Diagonal Matrix
However, any other Diagonal Matrix will have
the effect of enlarging a Vector in given axes.
For example, the following Diagonal Matrix:
Has the effect of stretching a Vector by a Scale
Factor of 2 in the x-Axis, 3 in the z-Axis and
reflecting the Vector in the y-Axis.

7. The Goal
By the end of this PowerPoint, we should be
able to understand and apply the idea of
Diagonalisation, using Eigenvalues and
Eigenvectors.

8. The Goal
The Matrix Point of View
By the end, we should be able to understand
how, given an n by n Matrix, A, we can say
that A is Diagonalisable if and only if there is
a Matrix, δ, that allows the following Matrix
to be Diagonal:
And why this knowledge is significant.

9. The Points of View
The Square Matrix, A, may be seen as a
Linear Operator, F, defined by:
Where X is a Column Vector.

10. The Points of View
Furthermore:
Represents the Linear Operator, F, relative to
the Basis, or Coordinate System, S, whose
Elements are the Columns of δ.

11. The Effects of a Coordinate System
If we are given A, an n by n Matrix of any
kind, then it is possible to interpret it as a
Linear Transformation in a given
Coordinate System of n-Dimensions.
For example:
Has the effect of 45 degree Anticlockwise Rotation, in
this case, on the Identity Matrix.

12. The Effects of a Coordinate System
However, it is theorised that it is possible to
represent this Linear Transformation as a
Diagonal Matrix within another, different
Coordinate System.
We define the effect upon a given Vector in
this new Coordinate System as:
The a scalar multiplication of the Vector
relative to all the axes by an unknown
Scale Factor, without affecting direction or
other properties.

13. The Effects of a Coordinate System
This process can be summarised by the
following definition:
Current New
Coordinate Coordinate
System System
Where:
A is the Transformation Matrix
v is a non-Zero Vector to be Transformed
λ is a Scalar in this new Coordinate System
that has the same effect on v as A.

14. The Effects of a Coordinate System
This process can be summarised by the
following definition:
Current New
Coordinate Coordinate
System System
Where:
: The Matrix is a Linear
Transformation upon the Vector .
: is the Scalar which results in the
same Transformation on as .

15. The Effects of a Coordinate System
This can be applied in the following example:
Matrix Vector
A
1
2

16. The Effects of a Coordinate System
This can be applied in the following example:
Matrix Vector
A
3
Thus, when:
4
A is equivalent to the Diagonal Matrix:
Which, as discussed
previously, has the effect of
enlarging the Vector by a
Scale Factor of 2 in each
Dimension.

17. Definitions
Thus, if is true, we call:
the Eigenvector
the Eigenvalue of corresponding to

18. Exceptions and Additions
• We do not count as an Eigenvector,
as for all values of λ.
• , however, is allowed as an accepted
Eigenvalue.
• If is a known Eigenvector of a Matrix,
then so is , for all non-Zero values of .
• If Vectors are both Eigenvectors of a
given Matrix, and both have the same
resultant Eigenvalue, then will also
be an Eigenvector of the Matrix.

19. Characteristic Polynomials
Establishing the Essentials
is an Eigenvalue for the Matrix , relative
to the Eigenvector . Is the Identity Matrix
of the same Dimensions as .
Thus: This is possible, as
Is invertible.

20. Characteristic Polynomials
Application of the Knowledge
What this, essentially, leads to is the finding
of all Eigenvalues and Eigenvectors of a
specific Matrix.
This is done by considering the Matrix, , in
addition to the Identity Matrix, .
We then multiply the Identity Matrix by the
unknown quantity, .

21. Characteristic Polynomials
Application of the Knowledge
Proceeding this, we then take the lots of
the Identity Matrix, and subtract the Matrix
from it.
We then take the Determinant of the result
which ends up as a Polynomial equation, in
order to find possible values of , the
Eigenvalues.
This can be exemplified by the following
example:

22. Characteristic Polynomials
Calculating Eigenvalues from a Matrix
To find the Eigenvalues of:
We must consider :

23. Characteristic Polynomials
Calculating Eigenvalues from a Matrix
Then, equals:
Which factorises to:
Therefore, the Eigenvalues of the Matrix are:

24. Characteristic Polynomials
Calculating Eigenvectors from the Values
With:
We need to solve for all given
values of .
This is done by solving a Homogeneous
System of Linear Equations. In other words,
we must turn into Echelon Form
and find the values of , which are
the Diagonals of the Matrix.

25. Characteristic Polynomials
Calculating Eigenvectors from the Values
For example, we will take from the
Previous example:

26. Characteristic Polynomials
Calculating Eigenvectors from the Values
For example, we will take from the
Previous example:
Therefore, the result is that:

27. Characteristic Polynomials
Calculating Eigenvectors from the Values
For example, we will take from the
Previous example:
Therefore, this is the set of general
Eigenvectors for the Eigenvalue of 4.

28. Diagonalisation
Mentioned earlier was the ultimate goal
of Diagonalisation; that is to say, finding a
Matrix, , such that the following can be
applied to a given Matrix, :
Where the result is a Diagonal Matrix.

29. Diagonalisation
There are a few rules that can be derived
from this:
Firstly, must be an Invertible Matrix, as the
Inverse is necessary to the calculation.
Secondly, the Eigenvectors of must
necessarily be Linearly Independent for this
to work.
Linear Independence will be covered later.

30. Diagonalisation
Eigenvectors, Eigenvalues & Diagonalisation
It turns out that the columns of the Matrix
are the Eigenvectors of the Matrix . This is
why they must be Linearly Independent, as
Matrix must be Invertible.
Furthermore, the Diagonal Entries of the
resultant Matrix are the Eigenvalues
associated with that Column of Eigenvectors.

31. Diagonalisation
Eigenvectors, Eigenvalues & Diagonalisation
For example, in the previous example, we can
create a Matrix from the Eigenvalues 4, 2
and 6, respectively.
It is as follows:

32. Diagonalisation
Eigenvectors, Eigenvalues & Diagonalisation
Furthermore, we can calculate that:

33. Diagonalisation
Eigenvectors, Eigenvalues & Diagonalisation
Thus, the Diagonalisation of can be created
by:
Solving this gives:
The Eigenvalues in the Order given!

34. Linear Independence
Introduction
This will be a brief section on Linear
Independence to enforce that the
Eigenvectors of must be Linearly
Independent for Diagonalisation to be
implemented.

35. Linear Independence
Linear Independency in x-Dimensions
The vectors are classified as a
Linearly Independent set of Vectors if the
following rule applies:
The only value of the Scalar, , which makes
the equation:
True is for all instances of

36. Linear Independence
Linear Independency in x-Dimensions
The vectors are classified as a
Linearly Independent set of Vectors if the
following rule applies:
The only value of the Scalar, , which makes
the equation:
True is for all instances of

37. Linear Independence
Linear Independency in x-Dimensions
If there are any non-zero values of at any
instance of within the equation, then this
set of Vectors, , is considered
Linearly Dependent.
It is to note that only one instance of at
non-zero is needed to make the dependence.

38. Linear Independence
Linear Independency in x-Dimensions
Therefore, if, say, at , the value of ,
then the vector set is Linearly Dependent.
But, if were to be omitted from the set,
given all other instances of were zero, then
the set would, therefore, become Linearly
Independent.

39. Linear Independence
Implications of Linear Independence
If the set of Vectors, is Linearly
Independent, then it is not possible to write
any of the Vectors in the set in terms of any
of the other Vectors within the same set.
Conversely, if a set of Vectors is Linearly
Dependent, then it is possible to write at
least one Vector in terms of at least one
other Vector.

40. Linear Independence
Implications of Linear Independence
For example, the Vector set of:
Is Linearly Dependent, as can be written
as:

41. Linear Independence
Implications of Linear Independence
For example, the Vector set of:
We can say, however, that this Vector set may
be considered as Linearly Independent if
were omitted from the set.

42. Linear Independence
Finding Linear Independency
The previous equation can be more usefully
written as:
More significantly, additionally, is the idea
that this can be translated into a
Homogeneous System of x Linear Equations,
where x is the Dimension quantity of the
System.

43. Linear Independence
Finding Linear Independency
The previous equation can be more usefully
written as:
More significantly, additionally, is the idea
that this can be translated into a
Homogeneous System of x Linear Equations,
where x is the Dimension quantity of the
System.

44. Linear Independence
Finding Linear Independency
Therefore, the Matrix of Coefficients, , is an
n by x Matrix, where n is the number of
Vectors in the System and x is the Dimensions
of the System.
The Columns of are equivalent to the
Vectors of the System, .

45. Linear Independence
Finding Linear Independency
To observe whether is Linearly
Independent or not, we need to put the
Matrix into Echelon Form.
If, when in Echelon Form, we can observe
that each Column of Unknowns has a Leading
Entry, then the set of Vectors are Linearly
Independent.

46. Linear Independence
Finding Linear Independency
If not, then the set of Vectors are Linearly
Dependent.
To find the Coefficients, we can put into
Reduced Echelon Form to consider the
general solutions.

47. Linear Independence
Finding Linear Independency: Example
Let us consider whether the following set of
Vectors are Linearly Independent:

48. Linear Independence
Finding Linear Independency: Example
These Vectors can be written in the following
form:

49. Linear Independence
Finding Linear Independency: Example
The following EROs put this Matrix into
Echelon Form:
As this Matrix has a leading entry for every
Column, we can conclude that the set of
Vectors is Linearly Independent.

50. Summary
Thus, to conclude:
is the formula for Eigenvectors
and Eigenvalues.
is a Matrix that has Eigenvectors and
Eigenvalues to be calculated.
is an Eigenvector of
is an Eigenvalue of , corresponding to

51. Summary
Thus, to conclude:
is the formula for Eigenvectors
and Eigenvalues.
Given and , we can find by Matrix
Multiplying and observing how many
times the result is, relative to .

52. Summary
Thus, to conclude:
is the Characteristic Polynomial
of . This is used to find the
general set of Eigenvalues of ,
and thus, its Eigenvectors.
This is done by finding the determinant of
and solving the resultant
Polynomial equation to isolate the
Eigenvalues.

53. Summary
Thus, to conclude:
is the Characteristic Polynomial
of . This is used to find the
general set of Eigenvalues of ,
and thus, its Eigenvectors.
Then, by Substituting the Eigenvalues back
into and reducing the Matrix to
Echelon Form, we can find the general set of
Eigenvectors for that Eigenvalue.

54. Summary
Thus, to conclude:
is the Diagonalisation of .
is a Matrix created from the Eigenvectors
of , where each Column is an Eigenvector.
In order for to exist, must necessarily
be Invertible, where the Eigenvectors of
are Linearly Independent.

55. Summary
Thus, to conclude:
is the Diagonalisation of .
is a Matrix created from the Eigenvectors
of , where each Column is an Eigenvector.
The resultant is a Diagonal Matrix
where the diagonal values are the
Eigenvalues in the same column as its
associated Eigenvectors.