This ppt includes the topic diagonalization of martices and with examples.

2. Presented By:
Amenah Gondal
Presented To:
Ma’am Mehwish
Class:
B.S.Ed (III)
Topic:
Diagonalization of Matrices

3. Definition:
An 𝑛 × 𝑛 matrix A is said to be diagonalizable if it is
similar to a diagonal matrix. That is, A is
diagonalizable if there exits an invertible matrix P
such that P-1AP is a diagonal matrix. The matrix P is
said to be diagonalize A.

4. The following procedure will diagonalize A:
I. Find the distinct eigenvalues of A.
II. Find n linearly independent eigenvectors
v1,v2,…..,vn corresponding to the eigenvalues of A
III. From the matrix P with 𝑣𝑖 as columns, i
=1,2,3,….,n.
IV. If P is orthogonal, then 𝑃 𝑇 𝐴𝑃 = 𝑃−1 𝐴𝑃 is the
required diagonal matrix. The diagonal entries
being the eigenvalues λ 1,λ 2, … … . . , λ 𝑛
corresponding to the eigenvectors
𝑣1, 𝑣2, … … … , 𝑣 𝑛 respectively.
Algorithm for Diagonalization:

5. Example:
Find a real orthogonal matrix P for which 𝐏−𝟏
𝐀𝐏 is
a diagonal matrix
A=
𝟏 𝟐
𝟐 𝟏
Solution:
Eigenvalues of A are given by
|A-λI|=0
Therefore,
1 − λ 2
2 1 − λ
=0

6. Or (1 − λ)2−4 = 0
Thus 1 − 2λ+λ2
− 4 = 0
λ2 − 2λ −3 = 0
(λ −3)(λ+1)=0
When λ=3 ,
A −3I=
1 − 3 2
2 1 − 3
=
−2 2
2 −2
𝐴 − 3𝐼 𝑣 = 0 ⇒
−2 2
2 −2
𝑥1
𝑥2
=
0
0
⇒ −2𝑥1 + 2𝑥2 = 0
and 2𝑥1 − 2𝑥2 = 0
λ= 3 , λ= −1

7. Both of these equations give 𝑥1 − 𝑥2 = 0. S.
𝑥1
𝑥2
=
𝑥1
𝑥1
=𝑥1
1
1
Thus 1 1 𝑇
is an eigenvector. We normalize it
to get
1
2
1
2
𝑇
.
When λ= −1 (A+I)v = 0 ⇒
2 2
2 2
𝑥1
𝑥2
=
0
0
i.e., 2𝑥1 + 2𝑥2 = 0 or 𝑥1 + 𝑥2 = 0 .
Therefore,
𝑥1
𝑥2
=
𝑥1
−𝑥1
=𝑥1
1
−1

8. Thus 1 −1 𝑇
is an eigenvector. We normalize it
to get
1
2
−1
2
𝑇
.
P =
𝑣1
𝑣2
=
1
2
1
2
1
2
−1
2
Then 𝑃−1
= 𝑃 𝑇
= 𝑃
and

9. 𝑃−1
𝐴𝑃 =
1
2
1
2
1
2
−1
2
1 2
2 1
1
2
1
2
1
2
−1
2
𝑃−1 𝐴𝑃 =
1
2
1
2
1
2
−1
2
3
2
−1
2
3
1
2
𝑷−𝟏
𝑨𝑷 =
𝟑 𝟎
𝟎 −𝟏