This document discusses eigen values, eigen vectors, and diagonalization of matrices. It defines eigen values as the roots of the characteristic equation of a matrix. Eigen vectors are non-zero vectors that satisfy AX=λX, where λ is the eigen value. Diagonalization is the process of transforming a matrix A into a diagonal matrix D using a similarity transformation with an invertible matrix P, such that D=P-1AP. The document provides examples to illustrate these concepts and lists various properties of eigen values and eigen vectors.

1. DEPARTMENT OF
PHYSICS
EIGEN VALUES,
EIGEN VECTORS, &
DIAGONALIZATION
Submitted by :-Vandana
(M.Sc-1st.)

2. CONTENTS
 Eigen values
 Eigen vectors
 Similarity Transformation
 Diagonalization of a matrix

3. EIGEN VALUES
To find eigen values first we find the
characteristic equation given by
| A- λI | = 0
Now the roots of the equation or values of λ
are called eigen values.

4. EXAMPLE
Let A be a square matrix of order 33
1 1 3
A= 1 5 1
3 1 1
According to Cayley Hamilton theorem
| A-λI | =0
Where λ is any scalar, and
I is the 3×3 unit matrix

5. The determinant of these matrix equated to
zero i.e.
1- λ 1 3
1 5-λ 1 =0
3 1 1-λ
On expanding the determinant, we get the
equation i.e.
λ³-7λ²+36=0
On solving these equations we get,
λ=-2,3,6
These are called eigen values.

6. PROPERTIES OF EIGEN VALUES
1. Any square matrix A and its transpose A՚
have the same eigen values.
2. The sum of the eigen values of a matrix is
equal to the trace of the matrix.
3. The product of the eigen values of a matrix A
is equal to the determinant of A.
4. The eigen values of a triangular matrix are
just the diagonal elements of the matrix.
5. The eigen values of an idempotent matrix
are either zero or unity.

7. 6. The sum of the eigen values of a matrix is
the sum of the elements of the principal
diagonal.
7. If λ is an eigen value of a matrix A then 1/λ
is the eigen value of Aˉ¹.
8. If λ is an eigen value of an orthogonal matrix
then 1/λ is also its eigen value.
9. If λ₁, λ₂, λ₃,…….,λո are the eigen values of a
matrix A then A ͫ has the eigen values λ₁ͫ ,λ₂ͫ
,……..,λոͫ.

8. EIGEN VECTOR
An eigen vector of a square matrix A is a non
zero vector X such that for some number λ,
we have,
AX=λX
AX-λIX=0
[A-λI]X=0
Where 0 represents the zero vector.

9. EXAMPLE
Now eigen vector corresponding to
λ=-2,3,6
For λ=-2
3x₁+x₂+3x₃=0
x₁+7x₂+x₃ =0
3x₁+x₂+3x₃=0
By solving these equations, we get
x₁=1, x₂=0, x₃=-1

10. For λ=3
-2x₁+x₂+3x₃=0
x₁+2x₂+x₃=0
3x₁+x₂-2x₃=0
By solving these equations, we get
X₁=1, x₂=-1, x₃=1

11. For λ=6
-5x₁+x₂+3x₃=0
x₁-x₂+x₃=0
3x₁+x₂-5x₃=0
By solving these equations, we get
x₁=1, x₂=2, x₃=1
Hence
λ=-2(1,0,-1)
λ=3(1,-1,1)
λ=6(1,2,1)
Hence corresponding to each eigen value there
exist an eigen vector.

12. PROPERTIES OF EIGEN VECTORS
1. The eigen vector X of a matrix A is not unique.
2. If λ₁, λ₂,……λո be distinct eigen values of an nᵡ n
matrix then corresponding to eigen vectors X₁,
X₂,……..Xո form a linearly independent set.
3. If two or more eigen values are equal it may or
may not be possible to get linearly independent
eigen vectors corresponding to the equal roots.
4. Two eigen vectors X₁ and X₂ are called
orthogonal vectors if X₁՚X₂=0.
5. Eigen vectors of a symmetric matrix
corresponding to different eigen values are
orthogonal.

13. NORMALISED FORM OF A VECTORS
To find normalised form of a we divide each
b
c
element by √a²+b²+c².
For example:-
Normalised form of 1 1/3
2 = 2/3
2 2/3

14. ORTHOGONAL VECTORS
Two vectors X and Y are said to be orthogonal if
X₁ᵀX₂=X₂ᵀx₁=0.
Let a matrix A is
1 0 -1
1 2 1
2 2 3
According to Cayley Hamilton theorem
|A-λI|=0

15. Characteristic equation is
1-λ 0 -1
1 2-λ 1 =0
2 2 3-λ
λ=1,2,3 are three distinct eigen values of A.
For λ=1
-x₃=0
x₁+x₂+x₃=0
2x₁+2x₂+2x₃=0

16. 1
X₁ = k -1
0
For λ=2
2
X₂ = K -1
-2
For λ=3

17. For λ=3
1
X₃=k -1
-2
X₁ᵀX₂=3≠0, X₂ᵀx₃=7≠0, X₃ᵀX₁=2≠0
Thus there are three distinct eigen vectors. So
X₁, X₂, X₃ are not orthogonal eigen vectors.

18. SIMILARITY TRANSFORMATION
Let A and B be two square matrices of order n.
Then B is said to be similar to A if there exists
a non-singular matrix P such that
B=Pˉ¹AP
This equation is called a similar transformation.

19. DIAGONALIZATION OF A MATRIX
Diagonalization of a matrix A is the process of
reduction of A to a diagonal form ‘D’. If A is
related to D by a similarity transformation such
that D=Pˉ¹AP then A is reduced to the diagonal
matrix D through model matrix P. D is also
called spectral matrix of A.

20. EXAMPLE
Let a matrix
6 -2 2
A= -2 3 -1
2 -1 3
By solving this matrix by Cayley Hamilton th ͫ
|A- λI|=0
We get
λ=2,2,8
These are called eigen values.

21. Eigen vector for λ=2
0
X₁= 1
1
Eigen vector for λ=2 again
1
X₂= 3
1

22. Eigen vector for λ=8
2
X₃= -1
1
Then power of a matrix is
0 1 2
P= 1 3 -1
1 1 1
4 1 -7
Pˉ¹= -1/6 -2 -2 2
-2 1 -1

23. Then
2 0 O
D = Pˉ¹AP= 0 2 0
0 0 1
Thus D is the diagonal matrix.

24. REFERENCES
● H.K Dass, Dr. Rama Verma,”Mathematical
Physics”, S. Chand & Company PVT.LTD., New
Delhi.
● Dr. B.S. Grewal”Higher Engineering
Mathematics”, Khanna Publishers, New Delhi,
2007.