Characteristic Vectors of Matrix

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

1 1 3 1 5 1 3 1 1

Solution : 

Now we have to multiply λ with unit matrix I.

To find roots let |A-λI| = 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 (λ-6) + 2 (λ-6) = 0

(λ+2) (λ-6) = 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

                             

=	-2   1   3 1   2   1 3   1   -2

 

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

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

                             

=	3   1   3 1   7   1 3   1   3

 

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

Substitute λ = 6 in the matrix A - λI

                             

=	-5   1   3 1   -1   1 3   1   -5

 

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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
• Diagonalization of Matrix
• Gauss Elimination Method
• Worksheet of matrices
  

Characteristic Vectors of Matrix to Characteristic Equation