In this page characteristic vectors of matrix1 we are going to see how to find characteristic equation of any matrix with detailed example.
Definition :
The eigen vector can be obtained from (A λI)X = 0. Here A is the given matrix λ is a scalar,I is the unit matrix and X is the columns matrix formed by the variables a,b and c.
Another name of characteristic Vector:
Characteristic vector are also known as latent vectors or Eigen vectors of a matrix.
Question 1 :
Determine the characteristic vector of the matrix

To find eigen vector first let us find characteristic roots of the given matrix.
Let A = 

The order of A is 3 x 3. So the unit matrix I = 

Now we have to multiply λ with unit matrix I.
λI = 

AλI= 

 

= 

= 

AλI= 
 
= (5λ)[(2λ) (5λ)  0]  0 [(0  0)] + 1 [0 (2 λ)] = (5λ)[ 10 + 2 λ  5 λ + λ²]  0 + 2 + λ = (5λ)[ 10  3 λ + λ²]  0 + 2 + λ = (5λ)[λ² 3 λ10] + 2 + λ = 5 λ²  15 λ  50  λ³ + 3 λ² + 10 λ + 2 + λ =  λ³ + 5 λ² + 3 λ²  15 λ + 10 λ + λ  50 + 2 =  λ³ + 8 λ²  4 λ  48 = λ³  8 λ² + 4 λ + 48 
To find roots let AλI = 0
λ³  8 λ² + 4 λ + 48 = 0
For solving this equation first let us do synthetic division.
By using synthetic division we have found one value of λ that is λ = 2.
Now we have to solve λ²  10 λ + 24 to get another two values. For that let us factorize diagonalization of matrix1
λ²  10 λ + 24 = 0
λ²  6 λ  4 λ + 24 = 0
λ (λ  6)  4 (λ  6) = 0
(λ  6) (λ  4) = 0
λ  6 = 0
λ = 6
λ  4 = 0
λ = 4
Therefore the characteristic roots (or) Eigen values are x = 2,4,6
Substitute λ = 2 in the matrix A  λI characteristic vectors of matrix1
= 

From this matrix we are going to form three linear equations using variables x,y and z.
7x + 0y + 1z = 0  (1)
0x + 0y + 0z = 0  (2)
1x + 0y + 7z = 0  (3)
By solving (1) and (3) we get the eigen vector characteristic vectors of matrix1
The eigen vector x = 

Substitute λ = 4 in the matrix A  λI
= 

From this matrix we are going to form three linear equations using variables x,y and z.
1x + 0y + 1z = 0  (4)
0x  6y + 0z = 0  (5)
1x + 0y + 4z = 0  (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y = 

Substitute λ = 6 in the matrix A  λI characteristic vectors of matrix1
= 

From this matrix we are going to form three linear equations using variables x,y and z.
1x + 0y + 1z = 0  (7)
0x + 8y + 0z = 0  (8)
1x + 0y  1z = 0  (9)
By solving (7) and (8) we get the eigen vector
The eigen vector z = 

Questions 
Solution 
characteristic vectors of matrix 1 characteristic vectors of matrix 1 characteristic vectors of matrix 1 characteristic vectors of matrix 1
Question 2 : Determine the characteristic vector of the matrix

 
Question 3 : Determine the characteristic vector of the matrix

 
Question 4 : Determine the characteristic vector of the matrix

 
Question 5 : Determine the characteristic vector of the matrix

characteristic vectors of matrix1characteristic vectors of matrix 1 