Characteristic Vectors of Matrix 4

In this page characteristic vectors of matrix 4 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 4 :

Determine the characteristic vector of the matrix

4 -20 -10 -2 10 4 6 -30 -13

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

Now we have to multiply λ with unit matrix I.

To find roots let |A-λI| = 0

   -λ³ + 1λ² + 2λ = 0

For solving this equation -λ from all the terms

-λ (λ² - 1λ - 2) = 0

-λ = 0 (or) λ² - 1 λ - 2 = 0

λ = 0       (λ+1) (λ-2) = 0 

               λ + 1 = 0       λ - 2 = 0

                  λ = - 1            λ = 2

Therefore the characteristic roots (or) Eigen values are x = 0,-1,2

Substitute λ = 0 in the matrix A - λI

                      

=	-4   -20   -10 -2   10   4 6   -30   -13

 

From this matrix we are going to form three linear equations using variables x,y and z.

4x - 20y - 10z = 0  ------ (1)

-2x + 10y + 4z = 0  ------ (2)

6x - 30y - 13z = 0  ------ (3)

By solving (1) and (2) we get the eigen vector

Substitute λ = -1 in the matrix A - λI

                      

=	5   -20   -10 -2   10   4 6   -30   -12

 

From this matrix we are going to form three linear equations using variables x,y and z.

5x - 20y - 10z = 0  ------ (4)

-2x + 10y + 4z = 0  ------ (5)

6x - 30y - 12z = 0  ------ (6)

By solving (4) and (5) we get the eigen vector  characteristic vectors of matrix4

Substitute λ = 2 in the matrix A - λI

                      

=	2   -20   -10 -2   8   4 6   -30   -15

 

From this matrix we are going to form three linear equations using variables x,y and z.

2x - 20y - 10z = 0  ------ (7)

-2x + 8y + 4z = 0  ------ (8)

6x - 30y - 15z = 0  ------ (9)

By solving (7) and (8) we get the eigen vector    characteristic vectors of matrix 4  characteristic vectors of matrix 4 characteristic vectors of matrix 4 characteristic vectors of matrix 4

---

---

Characteristic Vectors of Matrix4 to Characteristic Equation