In this page diagonalization of matrix 3 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 3 :
Diagonalize the following matrix

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= 

 

= 

= 

= (2λ)[ λ(1λ)  12 ]  2[2 λ  6]  3 [4(1)(1λ) ]
= (2λ)[ λ + λ²  12 ] + 4 λ + 12  3 [4+1λ ]
= (2λ)[ λ² λ  12 ] + 4 λ + 12  3 [3λ ]
= (2λ) [λ² λ  12 ] + 4 λ + 12 + 9 + 3 λ
= 2λ² + 2λ + 24  λ³ + λ² + 12 λ + 4 λ + 12 + 9 + 3 λ
=  λ³  λ² + 2λ + 12 λ + 4 λ + 3 λ + 24 + 12 + 9
=  λ³  λ² + 21λ + 45
= λ³ + λ²  21λ  45
To find roots let AλI = 0
λ³ + λ²  21λ  45 = 0 diagonalization of matrix 3
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  2 λ  15 to get another two values. For that let us factorize
λ²  2 λ  15 = 0
λ² + 3 λ  5 λ  15 = 0
λ (λ + 3)  5 (λ + 3) = 0
(λ  5) (λ + 3) = 0
λ  5 = 0
λ = 5
λ + 3 = 0
λ =  3
Therefore the characteristic roots (or) Eigen values are x = 3,3,5
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.
1x + 2y  3z = 0  (1)
2x + 4y  6z = 0  (2)
1x  2y + 3z = 0  (3)
By solving (1) and (3) we get the eigen vector
The eigen vector x = 

Substitute λ = 5 in the matrix A  λI Diagonalization of Matrix3
= 

From this matrix we are going to form three linear equations using variables x,y and z.
7x + 2y  3z = 0  (4)
2x  4y  6z = 0  (5)
1x  2y  5z = 0  (6)
By solving (4) and (5) we get the eigen vector Diagonalization of Matrix3
The eigen vector z = 

Let P = 

Eigen vectors of x and y are linearly dependent. So we cannot find diagonal matrix. diagonalization of matrix 3
Questions 
Solution 
Question 1 : Diagonalize the following matrix

 
Question 2 : Diagonalize the following matrix

 
Question 4 : Diagonalize the following matrix

 
Question 5 : Diagonalize the following matrix diagonalization of matrix 3 diagonalization of matrix 3

