This document provides an introduction to eigen values and eigen vectors. It defines eigen values as scalar values for which the equation Ax = λx has a non-trivial solution, where A is a matrix and x is an eigenvector. The characteristic equation is defined as det(A - λI) = 0, where the roots are the eigen values. Methods for determining eigen values and eigen vectors are described, including using the characteristic polynomial and characteristic matrix. Properties of eigen values are outlined, and the Cayley-Hamilton theorem is explained, which states that any matrix satisfies its own characteristic equation. An example is provided to demonstrate calculating eigen values and vectors.

1. EIGEN VALUES AND
EIGEN VECTORS
Continuous Assessment -1
Presented By: SatyabrataMondal
Department: CSE
UniversityRoll: 12000123082
Subject: MathematicsIA
Subject code: BS M101
Semester: 1st
College:Dr.B.C. Roy EngineeringCollege

2. INTRODUCTION
• If A is an n x n matrix and λ is a scalar for which Ax = λx has a
nontrivial solution x ∈ ℜⁿ, then λ is an eigenvalue of A and x is
a corresponding eigenvector of A.
◦
◦
Ax=λx=λIx
(A-λI)x=0
• The matrix (A-λI ) is called the characteristic matrix of a
where I is the Unit matrix.
◦
• The equation det (A-λI )= 0 is called characteristic
equationof A and the roots of this equation are called the
eigenvalues of the matrix A. The set of all eigenvectors is
called the eigenspace of A corresponding to λ. The set of all
eigenvalues of a is called spectrum of A.

3. Determining Eigenvalues
● Vector equation
◦ Ax = x (A-)x = 0
◦ A- is called the characteristic matrix
● Non-trivial solutions exist if and only if:
◦ This is called the characteristic equation
● Characteristic polynomial
◦ nth-order polynomial in 
◦ Roots are the eigenvalues {1, 2, …, n}
= 0
−
an1 an2 ann
a12
a1n
a2n
⁝
a11 −
a21 a22
⁝
−
⁝ ⋱
det(A − I) =

4. Eigenvalue Example
● Characteristic matrix


     3 − 4 −
0 1 
0
=
1− 2
− 4
3
2 
− 
1
A− I =
1
● Characteristic equation
A −I = (1−)(−4 − ) − (2)(3) = 2
+3 −10 = 0
● Eigenvalues: 1 = -5, 2 = 2

5. Eigenvalue Properties
● Eigenvalues of A and AT are equal
● Singular matrix has at least one zero eigenvalue
● Eigenvalues of A-1: 1/1, 1/2, …, 1/n
● Eigenvalues of diagonal and triangular matrices
are equal to the diagonal elements
● Trace n
j =1
● Determinant
Tr ( A) =   j
n
A =  j
j=1

6. Determining Eigenvectors
● Eigenvectors determined up to scalar
multiple
● Distinct eigenvalues
◦ Produce linearly independent eigenvectors
● Repeated eigenvalues
◦ Produce linearly dependent eigenvectors
◦ Procedure to determine eigenvectors more complex
(see text)
◦ Will demonstrate in Matlab
● First determine eigenvalues: {1, 2, …, n}
● Then determine eigenvector corresponding
to each eigenvalue:
(A − I)x = 0  (A − k I)xk = 0

7. Eigenvector Example
● Eigenvalues
● Determine eigenvectors: Ax = x
● Eigenvector for 1 = 2
 


 −3
1 1
3x1 + x2 = 0  0.9487 
● Eigenvector for 1 = -5
6x1 + 2x2 = 0
x =
−0.3162
or x =
 1 
 


 1
0.4472
2 2
x =
0.8944
or x =
2
3x1 −6x2 =0
−x1 +2x2 =0
2
3
 
− 4  = 2
2  1 = −5
A =
1

(1− )x1 + 2x2 = 0
3x1 −(4 + )x2 = 0
x1 + 2x2 = x1
3x1 − 4x2 = x2

8. Cayley-Hamilton Theorem



 
 
2 1 2 2 2 n
n 1 n 2 n n
● Let the characteristic polynomial of A be
 (λ)then,
φ ( λ ) = A - λ I
a 1 2 a 1 n
 a 1 1 - λ
 a a
= 
. . . . . . . . .
 a a
. . .
a - λ . . .
. . .
. . . a - λ

The characteristic equation
| A -λI|= 0

9. ● Verify Cayley – Hamilton theorem for
the matrix A= . Hence
compute A-1 .
●Solution:- The characteristic equation
of A is



2 
−1
1
−1
−1 2
−1 1 
2
(on simplification)
or λ3
−6λ2
+9λ−4 = 0
2−λ −1 1
A−λI = 0 i.e., −1 2−λ −1 = 0
1 −1 2−λ

10. 
22

−2
1

 
2
 
21
−1
=−21

6

1


5
6 −5 52 −1 1 2.2 −22 −21

6 −5
−1 2 22
−5 −1 −21

6

 
2
 
5
−1
 =−5 −5


2

1


1
−1
−1
2 2 6
−1 −1 −5
2 −1 12 −1 1 6 −5 5
A3
=A2
A=−5
A2
=−1
To verify Cayley – Hamilton theorem, we have to
show that A3 – 6A2 +9A – 4I = 0 … (1)

11.  A3 -6A2 +9A – 4I = 0
− 21
 22 − 22 − 21

= − 21 22 
 21 − 21 22 

6 

 5
− 5
− 5
 6 − 5 5 
6 
− 5
- 6 + 9  
2 


1
−1
−1
 2 −1 1 
 2
−1
 
0 1
0

-4 0
1 0 0
1
0

0
= 0

0
0 0 0
0 0 = 0
0
This verifies Cayley – Hamilton theorem

12. References
• Engineering Mathematics
• https://prezi.com
• https://byjus.com/maths/eigen-values/