In this page diagonalization of matrix 5 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 5 :
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= 

 

= 

= 

AλI= 
 
= (11λ)[(2λ)(6λ)20]+4[7(6λ)+50]7[2810(2λ)] = (11λ)[(2+λ)(6+λ)20]+4[427λ+50]7[28+20+10λ] = (11λ)[12+2λ+6λ+λ²20]+4[87λ]7[8+10λ] = (11λ)[λ²+8λ8]+3228λ+5670λ = 11λ²+8λ88λ³8λ²+88λ98λ+88 =  λ³+3λ²2λ = λ³+3λ²2λ = λ(λ²3λ²+2) 
To find roots let AλI = 0
λ(λ²3λ²+2) = 0
λ = 0
Now we have to solve λ²3λ²+2 to get another two values. For that let us factorize
λ²3λ²+2 = 0
λ²1λ2λ+2 = 0
λ(λ1)2(λ1) = 0
(λ1)(λ2) = 0
λ  1 = 0
λ = 1
λ  2 = 0
λ = 2
Therefore the characteristic roots (or) Eigen values are x = 0,1,2
Substitute λ = 0 in the matrix A  λI diagonalization of matrix 5
AλI= 

From this matrix we are going to form three linear equations using variables x,y and z.
11x  4y  7z = 0  (1)
7x  2y  5z = 0  (2)
10x  4y  6z = 0  (3)
By solving (1) and (2) we get the eigen vector
The eigen vector x = 

Substitute λ = 1 in the matrix A  λI
= 

From this matrix we are going to form three linear equations using variables x,y and z.
10x  4y  7z = 0  (4)
7x  3y  5z = 0  (5)
10x  4y  7z = 0  (6)
By solving (4) and (5) we get the eigen vector
The eigen vector y = 

Substitute λ = 2 in the matrix A  λI diagonalization of matrix 5
= 

From this matrix we are going to form three linear equations using variables x,y and z.
9x  4y  7z = 0  (7)
7x  4y  5z = 0  (8)
10x  4y  8z = 0  (9)
By solving (7) and (8) we get the eigen vector
The eigen vector z = 

Let P = 

The column of P are linearly independent eigen vectors of A . Therefore the diagonal matrix = diagonalization of matrix 5 

Questions 
Solution 
Question 1 : Diagonalize the following matrix

 
Question 2 : Diagonalize the following matrix

 
Question 3 : Diagonalize the following matrix

 
Question 4 : Diagonalize the following matrix diagonalization of matrix 5

