Characteristic Vectors of Matrix 1

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

5 0 1 0 -2 0 1 0 5

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 - 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

                             

=	7   0   1 0   0   0 1   0   7

 

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

Substitute λ = 4 in the matrix A - λI

                             

=	1   0   1 0   -6   0 1   0   4

 

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

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

                             

=	-1   0   1 0   -8   0 1   0   -1

 

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

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• Equality of matrices
• Operation on matrices
• Algebraic properties of matrices
• Addition properties
• Multiplication properties
• Minor of matrix
• Determinant of matrix
• Adjoint of matrix
• Inverse of matrix
• Solving linear equation by inversion method
• Solving linear equations of two unknowns by cramer method
• Solving linear equations of three unknowns by cramer method
• Rank of matrix
• Solve linear equation of three unknowns by rank method
• Linear dependence of vector
• Characteristic Equation of Matrix
• Characteristic Vector of Matrix
• Diagonalization of Matrix
• Gauss Elimination Method
• Worksheet of matrices
  

Characteristic Vectors of Matrix1 to Characteristic Equation