In this page characteristic vectors of matrix 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.
Example :
Determine the characteristic vector of the matrix

Solution :
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= 
 
= (1λ) [(5λ) (1λ)  1]1 [1(1λ)  3] + 3 [1  3 (5λ)] = (1λ) [55λλ+λ²1]  1 [1 λ  3] + 3[115+3λ] = (1λ) [λ²6λ+4]  1[2  λ] + 3[14+3λ] = λ²  6λ + 4  λ³ + 6λ²  4λ + 2 + λ  42 + 9λ =  λ³ + 7 λ²  10λ + 10λ + 6  42 =  λ³ + 7 λ²  36 
To find roots let AλI = 0
 λ³ + 7 λ²  36 = 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 = 3.
Now we have to solve λ²  4 λ  12 to get another two values.For that let us use factoring method.characteristic vectors of matrix
λ²  4 λ  12 = 0
λ²  6 λ + 2 λ  12 = 0
λ (λ6) + 2 (λ6) = 0
(λ+2) (λ6) = 0
λ + 2 = 0 λ  6 = 0
λ = 2 λ = 6
Therefore the characteristic roots are x = 3,2 and 6
Substitute λ = 3 in the matrix A  λI
= 

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

Substitute λ = 2 in the matrix A  λI characteristic vectors of matrix
= 

From this matrix we are going to form three linear equations using variables x,y and z.
3x + 1y + 3z = 0  (4)
1x + 7y + 1z = 0  (5)
3x + 1y + 3z = 0  (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y = 

Substitute λ = 6 in the matrix A  λI
= 

The eigen vector z = 

Let P = 

The column of P are linearly independent eigen vectors of A . Therefore the diagonal matrix = 

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

 
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 matrix 
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