Linear transformations on planeEigen values
Markov Matrices
Eigenvalues, eigenvectors and applications
Dr. D. Sukumar
Department of MathematicsIndian Institute of Technology Hyderabad
Recent Trends in Applied Sciences with Engineering ApplicationsJune 27-29, 2013
Department of Applied ScienceGovernment Engineering College,Kozhikode, Kerala
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Maps which preserve
Origin
lines passing through origin
parallelograms with one corner as origin
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Outline
1 Linear transformations on planeTypical ExamplesProperties
2 Eigen valuesEigen value and eigen vector
3 Markov MatricesFormationInterpretationProperties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Rotation
(0 −11 0
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Rotation
(0 −11 0
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Reflection
(1 00 −1
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Reflection
(1 00 −1
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Expansion
(2 00 2
)Compression
(1/2 0
0 1/2
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Expansion
(2 00 2
)Compression
(1/2 0
0 1/2
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Expansion
(2 00 2
)Compression
(1/2 0
0 1/2
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Multi-scaling or Stretching
(2 00 3
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Multi-scaling or Stretching
(2 00 3
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Projection
(1 00 0
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Projection
(1 00 0
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Shear transformation
(1 10 1
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Shear transformation
(1 10 1
)
-1 1 2-1
1
2
x
y
x
y
-1 1 2-1
1
2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Outline
1 Linear transformations on planeTypical ExamplesProperties
2 Eigen valuesEigen value and eigen vector
3 Markov MatricesFormationInterpretationProperties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Properties
Area
Eigen vectors
Eigen values
Determinant
Diagonalizable
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Properties
Area
Eigen vectors
Eigen values
Determinant
Diagonalizable
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Properties
Area
Eigen vectors
Eigen values
Determinant
Diagonalizable
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Properties
Area
Eigen vectors
Eigen values
Determinant
Diagonalizable
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Properties
Area
Eigen vectors
Eigen values
Determinant
Diagonalizable
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
Typical ExamplesProperties
Table of properties
Map Area Fixed Dir Scale in FD Det DiagonableEigenvector Eigenvalue
Rotation 1 NO NO 1 NO
Reflection 1 x-axis, y -axis 1,-1 -1 Yes
Expansion 4 x-axis, y -axis 2, 2 4 YesCompression 1/4 x-axis, y -axis 1/2,1/2 1/4 Yes
Multi-scaling 6 x-axis, y -axis 2,3 6 Yes
Projection 0 x-axis, y -axis 1,0 0 Yes
Shear 1 x-axis 1 1 NO
Table: Properties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Problem
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
Purpose
Showing all steps of this process using linear algebra
Mainly using eigenvalues and eigenvectors
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Problem
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
Purpose
Showing all steps of this process using linear algebra
Mainly using eigenvalues and eigenvectors
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Problem
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
Purpose
Showing all steps of this process using linear algebra
Mainly using eigenvalues and eigenvectors
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Problem
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
Purpose
Showing all steps of this process using linear algebra
Mainly using eigenvalues and eigenvectors
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Outline
1 Linear transformations on planeTypical ExamplesProperties
2 Eigen valuesEigen value and eigen vector
3 Markov MatricesFormationInterpretationProperties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Eigen value
Let A a be square matrix.
Eigen values of A are solutions or roots of
det(A− λI ) = 0.
IfAx = λx or (A− λI )x = 0,
for a non-zero vector x then
λ is an eigenvalue of A andx is an eigenvector corresponding to the eigenvalue λ.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Eigen value
Let A a be square matrix.
Eigen values of A are solutions or roots of
det(A− λI ) = 0.
IfAx = λx or (A− λI )x = 0,
for a non-zero vector x then
λ is an eigenvalue of A andx is an eigenvector corresponding to the eigenvalue λ.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Eigen value
Let A a be square matrix.
Eigen values of A are solutions or roots of
det(A− λI ) = 0.
IfAx = λx or (A− λI )x = 0,
for a non-zero vector x then
λ is an eigenvalue of A and
x is an eigenvector corresponding to the eigenvalue λ.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Eigen value
Let A a be square matrix.
Eigen values of A are solutions or roots of
det(A− λI ) = 0.
IfAx = λx or (A− λI )x = 0,
for a non-zero vector x then
λ is an eigenvalue of A andx is an eigenvector corresponding to the eigenvalue λ.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Example
Consider the matrix A =
[2 53 0
].
Example (Eigen value)
A− λI =
[2 53 0
]− λ
[1 00 1
]=
[2− λ 5
3 −λ
]det(A− λI ) = (2− λ)(−λ)− (3× 5) = λ2 − 2λ− 15The roots of the polynomial are the eigen values: -3 and 5.
Example (When λ = −3)[2− (−3) 5
3 −(−3)
](x1x2
)=
(00
)Eigen vector
(−11
)Example (When λ = 5)[
2− 5 53 −5
](x1x2
)=
(00
)Eigen vector
(53
)
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Example
Consider the matrix A =
[2 53 0
].
Example (Eigen value)
A− λI =
[2 53 0
]− λ
[1 00 1
]=
[2− λ 5
3 −λ
]
det(A− λI ) = (2− λ)(−λ)− (3× 5) = λ2 − 2λ− 15The roots of the polynomial are the eigen values: -3 and 5.
Example (When λ = −3)[2− (−3) 5
3 −(−3)
](x1x2
)=
(00
)Eigen vector
(−11
)Example (When λ = 5)[
2− 5 53 −5
](x1x2
)=
(00
)Eigen vector
(53
)
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Example
Consider the matrix A =
[2 53 0
].
Example (Eigen value)
A− λI =
[2 53 0
]− λ
[1 00 1
]=
[2− λ 5
3 −λ
]det(A− λI ) = (2− λ)(−λ)− (3× 5) = λ2 − 2λ− 15
The roots of the polynomial are the eigen values: -3 and 5.
Example (When λ = −3)[2− (−3) 5
3 −(−3)
](x1x2
)=
(00
)Eigen vector
(−11
)Example (When λ = 5)[
2− 5 53 −5
](x1x2
)=
(00
)Eigen vector
(53
)
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Example
Consider the matrix A =
[2 53 0
].
Example (Eigen value)
A− λI =
[2 53 0
]− λ
[1 00 1
]=
[2− λ 5
3 −λ
]det(A− λI ) = (2− λ)(−λ)− (3× 5) = λ2 − 2λ− 15The roots of the polynomial are the eigen values: -3 and 5.
Example (When λ = −3)[2− (−3) 5
3 −(−3)
](x1x2
)=
(00
)Eigen vector
(−11
)Example (When λ = 5)[
2− 5 53 −5
](x1x2
)=
(00
)Eigen vector
(53
)
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Example
Consider the matrix A =
[2 53 0
].
Example (Eigen value)
A− λI =
[2 53 0
]− λ
[1 00 1
]=
[2− λ 5
3 −λ
]det(A− λI ) = (2− λ)(−λ)− (3× 5) = λ2 − 2λ− 15The roots of the polynomial are the eigen values: -3 and 5.
Example (When λ = −3)[2− (−3) 5
3 −(−3)
](x1x2
)=
(00
)Eigen vector
(−11
)
Example (When λ = 5)[2− 5 5
3 −5
](x1x2
)=
(00
)Eigen vector
(53
)
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
Example
Consider the matrix A =
[2 53 0
].
Example (Eigen value)
A− λI =
[2 53 0
]− λ
[1 00 1
]=
[2− λ 5
3 −λ
]det(A− λI ) = (2− λ)(−λ)− (3× 5) = λ2 − 2λ− 15The roots of the polynomial are the eigen values: -3 and 5.
Example (When λ = −3)[2− (−3) 5
3 −(−3)
](x1x2
)=
(00
)Eigen vector
(−11
)Example (When λ = 5)[
2− 5 53 −5
](x1x2
)=
(00
)Eigen vector
(53
)Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
For the matrix A =
[2 53 0
]the eigenvalues are -3 and 5
the eigenvector corresponding to -3 is
[−11
][
2 53 0
] [−11
]= −3
[−11
]
the eigenvector corresponding to 5 is
[53
][
2 53 0
] [53
]=
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
For the matrix A =
[2 53 0
]the eigenvalues are -3 and 5
the eigenvector corresponding to -3 is
[−11
][
2 53 0
] [−11
]=
[3−3
]= −3
[−11
]
the eigenvector corresponding to 5 is
[53
][
2 53 0
] [53
]=
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
For the matrix A =
[2 53 0
]the eigenvalues are -3 and 5
the eigenvector corresponding to -3 is
[−11
][
2 53 0
] [−11
]= −3
[−11
]
the eigenvector corresponding to 5 is
[53
][
2 53 0
] [53
]=
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
For the matrix A =
[2 53 0
]the eigenvalues are -3 and 5
the eigenvector corresponding to -3 is
[−11
][
2 53 0
] [−11
]= −3
[−11
]
the eigenvector corresponding to 5 is
[53
][
2 53 0
] [53
]=
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov MatricesEigen value and eigen vector
For the matrix A =
[2 53 0
]the eigenvalues are -3 and 5
the eigenvector corresponding to -3 is
[−11
][
2 53 0
] [−11
]= −3
[−11
]
the eigenvector corresponding to 5 is
[53
][
2 53 0
] [53
]= 5
[53
]
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Outline
1 Linear transformations on planeTypical ExamplesProperties
2 Eigen valuesEigen value and eigen vector
3 Markov MatricesFormationInterpretationProperties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou.
For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging: How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200Normal mark: , , , ,
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou. For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging: How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200Normal mark: , , , ,
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou. For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging:
How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200Normal mark: , , , ,
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou. For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging: How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200
Normal mark: , , , ,
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou. For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging: How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200Normal mark: 100/200, 20/200,50/200 , 30/200, 0/200
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou. For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging: How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200Normal mark: 0.50, 0.1, 0.25, 0.15, 0.0
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
5 Volunteers
1 Data: Each one should give your marking for each one ofyou. For example: Isha: (I)100, (J)20, (K)50, (L)30, (M)0
2 Normalizing or averaging: How much you have given from thetotal marking
Total mark: 100+20+50+30+0=200Normal mark: 0.50, 0.1, 0.25, 0.15, 0.0
3 Arranging: Writing them in a matrix form0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Isha Jose Kumar Latha ManiIsha 0.5 0.8 0.0 0.4 0.2Jose 0.1 0.32 0.0 0.0 0.2
Kumar 0.25 0.24 1.0 0.0 0.2Latha 0.15 0.16 0.0 0.4 0.2Mani 0.0 0.2 0.0 0.2 0.2
↓ ↓ ↓ ↓ ↓sum is 1 1 1 1 1
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Markov Matrix
M =
m11 m12 · · · m1n
m21 m22 m2n...
......
mn1 mn2 mnn
n×n
Definition (Markov Matrix)
A real n × n matrix is called a Markov matrix if
1 Each entry is non-negative: mij ≥ 0 for all1 ≤ i ≤ n, 1 ≤ j ≤ n.
2 Each column sum is 1:∑n
i=1mij = 1 for each 1 ≤ j ≤ n.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Markov Matrix
M =
m11 m12 · · · m1n
m21 m22 m2n...
......
mn1 mn2 mnn
n×n
Definition (Markov Matrix)
A real n × n matrix is called a Markov matrix if
1 Each entry is non-negative: mij ≥ 0 for all1 ≤ i ≤ n, 1 ≤ j ≤ n.
2 Each column sum is 1:∑n
i=1mij = 1 for each 1 ≤ j ≤ n.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Outline
1 Linear transformations on planeTypical ExamplesProperties
2 Eigen valuesEigen value and eigen vector
3 Markov MatricesFormationInterpretationProperties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.
0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.
0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
10000
=
0.50.1
0.250.15
0
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.
0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
01000
=
0.8
0.320.240.160.2
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.
0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
00001
=
0.20.20.20.20.2
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.
0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
0.20.20.20.20.2
=
0.3800.1240.3380.1820.120
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
0.20.40.10.20.1
=
0.3200.0720.3940.1860.100
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Interpretation
We have formed a Markov matrix from individual opinions.
If Isha’s opinion is full value then Isha is better than others.
If Jose’s opinion is full value then Isha is better than others.
If Mani’s opinion is full value then all are equal.
If all opinions are equal value then Isha is better than others.
If opinion has different value then Kumar is better than others.0.5 0.8 0.0 0.4 0.20.1 0.32 0.0 0.0 0.2
0.25 0.24 1.0 0.0 0.20.15 0.16 0.0 0.4 0.20.0 0.2 0.0 0.2 0.2
x1x2x3x3x4
=
x1x2x3x4x5
When the opinion values matches with the conclusion value?.For which opinion value we will get the same conclusion value.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
For a Markov matrix M, we need a stable opinion value x .
Find a vector x such that
Mx = x
For M, find an eigenvector x corresponding to eigenvalue 1.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
For a Markov matrix M, we need a stable opinion value x .
Find a vector x such that
Mx = x
For M, find an eigenvector x corresponding to eigenvalue 1.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
For a Markov matrix M, we need a stable opinion value x .
Find a vector x such that
Mx = x
For M, find an eigenvector x corresponding to eigenvalue 1.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
For a Markov matrix M, we need a stable opinion value x .
Find a vector x such that
Mx = x
For M, find an eigenvector x corresponding to eigenvalue 1.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Big Problem
Getting a common opinion from individual opinion
From individual preference to common preference
For a Markov matrix M, we need a stable opinion value x .
Find a vector x such that
Mx = x
For M, find an eigenvector x corresponding to eigenvalue 1.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Outline
1 Linear transformations on planeTypical ExamplesProperties
2 Eigen valuesEigen value and eigen vector
3 Markov MatricesFormationInterpretationProperties
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Properties
Theorem
For every Markov matrix 1 is surely an eigenvalue.
Proof.
For a Markov matrix M every column sum is 1. Consider the
transpose matrix MT . Now its row sum is 1. Let e =
1...1
MT e = e
1 is an eigenvalue of MT with e as its eigenvector.
We know det(A) = det(AT ), so det(M − λI ) = det(MT − λI )Hence 1 is an eigenvalue of M
But eigen vectors of A and AT need not be same.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Properties
Theorem
For every Markov matrix 1 is surely an eigenvalue.
Proof.
For a Markov matrix M every column sum is 1. Consider the
transpose matrix MT . Now its row sum is 1. Let e =
1...1
MT e = e
1 is an eigenvalue of MT with e as its eigenvector.
We know det(A) = det(AT ), so det(M − λI ) = det(MT − λI )Hence 1 is an eigenvalue of M
But eigen vectors of A and AT need not be same.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Properties
Theorem
For every Markov matrix 1 is surely an eigenvalue.
Proof.
For a Markov matrix M every column sum is 1. Consider the
transpose matrix MT . Now its row sum is 1. Let e =
1...1
MT e = e
1 is an eigenvalue of MT with e as its eigenvector.
We know det(A) = det(AT ), so det(M − λI ) = det(MT − λI )Hence 1 is an eigenvalue of M
But eigen vectors of A and AT need not be same.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Properties
Theorem
For every Markov matrix 1 is surely an eigenvalue.
Proof.
For a Markov matrix M every column sum is 1. Consider the
transpose matrix MT . Now its row sum is 1. Let e =
1...1
MT e = e
1 is an eigenvalue of MT with e as its eigenvector.
We know det(A) = det(AT ), so det(M − λI ) = det(MT − λI )
Hence 1 is an eigenvalue of M
But eigen vectors of A and AT need not be same.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Properties
Theorem
For every Markov matrix 1 is surely an eigenvalue.
Proof.
For a Markov matrix M every column sum is 1. Consider the
transpose matrix MT . Now its row sum is 1. Let e =
1...1
MT e = e
1 is an eigenvalue of MT with e as its eigenvector.
We know det(A) = det(AT ), so det(M − λI ) = det(MT − λI )Hence 1 is an eigenvalue of M
But eigen vectors of A and AT need not be same.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Properties
Theorem
For every Markov matrix 1 is surely an eigenvalue.
Proof.
For a Markov matrix M every column sum is 1. Consider the
transpose matrix MT . Now its row sum is 1. Let e =
1...1
MT e = e
1 is an eigenvalue of MT with e as its eigenvector.
We know det(A) = det(AT ), so det(M − λI ) = det(MT − λI )Hence 1 is an eigenvalue of M
But eigen vectors of A and AT need not be same.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Further Properties
The previous result guaranteed the eigenvalue 1, but there is aproblem if it has more eigenvectors corresponding to eigenvalue 1.
Theorem
For a Markov matrix M with all entries mij > 0,
1 1 is always an eigenvalue.
2 Other eigenvalues have magnitude less than 1 ( |λ| < 1 ).
3 There is an eigenvector corresponding to the eigenvalue 1.
4 This eigenvector has all positive entries with sum 1.
5 Such an eigenvector is unique.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Further Properties
The previous result guaranteed the eigenvalue 1, but there is aproblem if it has more eigenvectors corresponding to eigenvalue 1.
Theorem
For a Markov matrix M with all entries mij > 0,
1 1 is always an eigenvalue.
2 Other eigenvalues have magnitude less than 1 ( |λ| < 1 ).
3 There is an eigenvector corresponding to the eigenvalue 1.
4 This eigenvector has all positive entries with sum 1.
5 Such an eigenvector is unique.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Further Properties
The previous result guaranteed the eigenvalue 1, but there is aproblem if it has more eigenvectors corresponding to eigenvalue 1.
Theorem
For a Markov matrix M with all entries mij > 0,
1 1 is always an eigenvalue.
2 Other eigenvalues have magnitude less than 1 ( |λ| < 1 ).
3 There is an eigenvector corresponding to the eigenvalue 1.
4 This eigenvector has all positive entries with sum 1.
5 Such an eigenvector is unique.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Further Properties
The previous result guaranteed the eigenvalue 1, but there is aproblem if it has more eigenvectors corresponding to eigenvalue 1.
Theorem
For a Markov matrix M with all entries mij > 0,
1 1 is always an eigenvalue.
2 Other eigenvalues have magnitude less than 1 ( |λ| < 1 ).
3 There is an eigenvector corresponding to the eigenvalue 1.
4 This eigenvector has all positive entries with sum 1.
5 Such an eigenvector is unique.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Further Properties
The previous result guaranteed the eigenvalue 1, but there is aproblem if it has more eigenvectors corresponding to eigenvalue 1.
Theorem
For a Markov matrix M with all entries mij > 0,
1 1 is always an eigenvalue.
2 Other eigenvalues have magnitude less than 1 ( |λ| < 1 ).
3 There is an eigenvector corresponding to the eigenvalue 1.
4 This eigenvector has all positive entries with sum 1.
5 Such an eigenvector is unique.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Further Properties
The previous result guaranteed the eigenvalue 1, but there is aproblem if it has more eigenvectors corresponding to eigenvalue 1.
Theorem
For a Markov matrix M with all entries mij > 0,
1 1 is always an eigenvalue.
2 Other eigenvalues have magnitude less than 1 ( |λ| < 1 ).
3 There is an eigenvector corresponding to the eigenvalue 1.
4 This eigenvector has all positive entries with sum 1.
5 Such an eigenvector is unique.
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Finding the stable opinion
Now we know that every Markov matrix with positive entries has aunique eigenvector corresponding to the eigenvalue λ = 1.
M − λI =
m11 m12 · · · m1n
m21 m22 m2n...
......
mn1 mn2 mnn
n×n
− λ
1 0 0 00 1 0 0...
.... . .
...0 0 0 1
n×n
(M − λI )x =
m11 − λ m12 · · · m1n
m21 m22 − λ m2n...
......
mn1 mn2 mnn − λ
x1x2...vn
=
00...0
Solve (M − λI )x = 0 by Gauss Elimination Process
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Finding the stable opinion
Now we know that every Markov matrix with positive entries has aunique eigenvector corresponding to the eigenvalue λ = 1.
M − λI =
m11 m12 · · · m1n
m21 m22 m2n...
......
mn1 mn2 mnn
n×n
− λ
1 0 0 00 1 0 0...
.... . .
...0 0 0 1
n×n
(M − λI )x =
m11 − λ m12 · · · m1n
m21 m22 − λ m2n...
......
mn1 mn2 mnn − λ
x1x2...vn
=
00...0
Solve (M − λI )x = 0 by Gauss Elimination Process
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Finding the stable opinion
Now we know that every Markov matrix with positive entries has aunique eigenvector corresponding to the eigenvalue λ = 1.
M − λI =
m11 m12 · · · m1n
m21 m22 m2n...
......
mn1 mn2 mnn
n×n
− λ
1 0 0 00 1 0 0...
.... . .
...0 0 0 1
n×n
(M − λI )x =
m11 − λ m12 · · · m1n
m21 m22 − λ m2n...
......
mn1 mn2 mnn − λ
x1x2...vn
=
00...0
Solve (M − λI )x = 0 by Gauss Elimination Process
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Finding the stable opinion
Now we know that every Markov matrix with positive entries has aunique eigenvector corresponding to the eigenvalue λ = 1.
M − λI =
m11 m12 · · · m1n
m21 m22 m2n...
......
mn1 mn2 mnn
n×n
− λ
1 0 0 00 1 0 0...
.... . .
...0 0 0 1
n×n
(M − λI )x =
m11 − λ m12 · · · m1n
m21 m22 − λ m2n...
......
mn1 mn2 mnn − λ
x1x2...vn
=
00...0
Solve (M − λI )x = 0 by Gauss Elimination Process
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Other Views
It is a simpler version of page-rank.
It represents transition matrix and steady state.
Markov process in input-output business models.
· · ·
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Other Views
It is a simpler version of page-rank.
It represents transition matrix and steady state.
Markov process in input-output business models.
· · ·
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Other Views
It is a simpler version of page-rank.
It represents transition matrix and steady state.
Markov process in input-output business models.
· · ·
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Other Views
It is a simpler version of page-rank.
It represents transition matrix and steady state.
Markov process in input-output business models.
· · ·
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Questions
Thank you
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
Questions
Thank you
Dr. D. Sukumar (IITH) Eigenvalues
Linear transformations on planeEigen values
Markov Matrices
FormationInterpretationProperties
You know better
In aeronautical engineering eigenvalues may determine whetherthe flow over a wing is laminar or turbulent.
In electrical engineering they may determine the frequencyresponse of an amplifier or the reliability of a nationalpower system.
In structural mechanics eigenvalues may determine whether anautomobile is too noisy or whether a building willcollapse in an earth-quake.
In probability they may determine the rate of convergence of aMarkov process.
In ecology they may determine whether a food web will settleinto a steady equilibrium.
In numerical analysis they may determine whether a discretizationof a differential equation will get the right answer orhow fast a conjugate gradient iteration will converge.
Dr. D. Sukumar (IITH) Eigenvalues
