Diagonalization of Matrix 2

In this page diagonalization of matrix 2 we are going to see how to diagonalize a matrix.

Definition :

A square matrix of order n is diagonalizable if it is having linearly independent eigen values.

We can say that the given matrix is diagonalizable if it is alike to the diagonal matrix. Then there exists a non singular matrix P such that P⁻¹ AP = D where D is a diagonal matrix.

Question 2 :

Diagonalize the following matrix

1 1 3 1 5 1 3 1 1

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.diagonalization of matrix 2

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 

   λ² - 9 λ + 18 = 0

λ² - 3 λ - 6 λ + 18 = 0

λ (λ - 3) - 6 (λ - 3) = 0

(λ - 6) (λ - 3) = 0

 λ - 6 = 0

 λ = 6

 λ - 3 = 0

 λ = 3

Therefore the characteristic roots (or) Eigen values are x = -2,3,6

Substitute λ = -2 in the matrix A - λI

=	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  ------ (1)

1x + 7y + 1z = 0  ------ (2)

3x + 1y + 3z = 0  ------ (3)

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

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  ------ (4)

1x + 2y + 1z = 0  ------ (5)

3x + 1y - 2z = 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

 

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

-5x + 1y + 3z = 0  ------ (7)

1x - 1y + 1z = 0  ------ (8)

3x + 1y - 5z = 0  ------ (9)

By solving (7) and (8) we get the eigen vector diagonalization of matrix 2

---

---

• Types of matrices
• Equality of matrices
• Operation on matrices
• Algebraic properties of matrices
• Multiplication properties
• Minor of a matrix
• Practice questions of minor of matrix
• Adjoint of matrix
• Determinant of matrix
• Inverse of a matrix
• Practice questions of inverse of matrix
• Inversion method
• Inversion method worksheets
• Cramer's Rule for 3 equations
• Cramer's Rule for 2 equations
• Solving 2 equations using Cramer's method
• Rank Method in Matrix
• Rank of a matrix
• Solving using rank method
• Linear dependence of vectors
• Linear dependence of vectors in rank method
• Characteristic Equation of matrix
• Characteristic vector of matrix
• Diagonalization of matrix
• Gauss elimination method
  
• Matrix calculator