This document discusses diagonalization of matrices. It defines similarity of matrices and notes that similar matrices have the same characteristic polynomial and eigenvalues. It then discusses diagonalizing matrices by finding the eigenvalues and corresponding eigenvectors, constructing a change of basis matrix P from the eigenvectors, and constructing a diagonal matrix D from the eigenvalues. It provides examples of diagonalizing matrices with real and complex eigenvalues.

1. 5
5.3
© 2012 Pearson Education, Inc.
Eigenvalues and Eigenvectors
DIAGONALIZATION

2. Slide 5.2- 2© 2012 Pearson Education, Inc.
SIMILARITY
 If A and B are nxn matrices, then A is similar to
B if there is an invertible matrix P such that
 P-1AP = B, or, equivalently
 A = PBP-1
 Changing A into B is called a similarity
transformation.

3. Slide 5.2- 3© 2012 Pearson Education, Inc.
SIMILARITY
 Theorem 4: If nxn matrices A and B are similar,
then they have the same characteristic polynomial
and hence the same eigenvalues (with the same
multiplicities).

4. Powers of a Diagonal Matrix
 Diagonal matrix – entries = 0 off diagonal
 Powers of D  diagonal with diagonal
entries powers of entries in D
 𝐷 =
𝑎 0
0 𝑏
, 𝐷 𝑛
=
𝑎 𝑛
0
0 𝑏 𝑛
Slide 5.3- 4© 2012 Pearson Education, Inc.

5. Slide 5.3- 5© 2012 Pearson Education, Inc.
DIAGONALIZATION
 Special Case: B is diagonal matrix  D
 Then A is “diagonalizable”
 Useful for computing powers of A
Example 1: Find Ak for 𝐴 =
7 2
−4 1
= PDP-1
𝑃 =
1 1
−1 −2
, 𝐷 =
5 0
0 3

6. Diagonalization Process – 4 steps
1. Find eigenvalues
2. Find corresponding eigenvectors
3. Construct P from eigenvectors
4. Construct D from eigenvalues
 Same order
Slide 5.3- 6© 2012 Pearson Education, Inc.

7. Slide 5.3- 7© 2012 Pearson Education, Inc.
DIAGONALIZING MATRICES - Example
Example: Diagonalize the following matrix, if
possible. 1 3 3
3 5 3
3 3 1
A
 
    
 
  

8. DIAGONALIZING MATRICES - Example
Slide 5.3- 8© 2012 Pearson Education, Inc.
Example: Diagonalize the following matrix, if
possible.
𝐴 =
2 4 3
−4 −6 −3
3 3 1
, with λ=1, -2 (multiplicity of 2)

9. Slide 5.3- 9© 2012 Pearson Education, Inc.
DIAGONALIZING MATRICES
Theorem 6: An nxn matrix with n distinct
eigenvalues is diagonalizable.

10. Diagonalizing Matrices
Theorem 7: Let A be an nxn matrix whose distinct
eigenvalues are λ1, …, λp.
a. For 1 ≤ k ≤ p, the dimension of the eigenspace
for λk ≤ multiplicity of the eigenvalue λk.
b. A diagonalizable iff sum of dim of each distinct
eigenspaces = n, which happens iff dim each
eigenspace = mult of eigenvalue
c. If A is diagonalizable, collection of all
eigenvectors form eigenvector basis for Rn
Slide 5.3- 10© 2012 Pearson Education, Inc.

11. Diagonalization - Example
Diagonalize 𝐴 =
5 0 0 0
0 5 0 0
1 4 −3 0
−1 −2 0 −3
Slide 5.3- 11© 2012 Pearson Education, Inc.

12. Complex Eigenvalues – Review Complex #s
 𝑖 = −1, i2 = -1
 Complex: a + bi
 a is real part
 bi is imaginary part
 Add: (a + bi) + (c + di) = (a + c) + (b + d)I
 Multiplication: (a + bi)(c + di)
= ac + adi + cbi + bdi2
= (ac – bd) + (bc + ad)i
 Conjugate 𝑎 + 𝑏𝑖 = a – bi
 (a + bi)(a – bi) = a2 – b2i2 = a2 + b2

13. Complex Eigenvalues – Purely Imaginary Ex
 Find eigenvalues and eigenvectors for
𝐴 =
0 −1
1 0
Slide 5.3- 13© 2012 Pearson Education, Inc.

14. Complex Eigenvalues - Example
 Find eigenvalues and eigenvectors for
𝐴 =
.5 −.6
.75 1.1
Slide 5.3- 14© 2012 Pearson Education, Inc.

15. Conjugates of Vectors & Matrices
 Conjugate each entry
 𝑎𝑏 = 𝑎 𝑏, regardless or scalar, vector, matrix
 𝐴𝐱 = 𝜆𝐱 (eigen relationship)
 𝐴𝐱 = 𝜆𝐱 (conjugate both sides)
 𝐴 𝐱 = 𝜆 𝐱 (A is real, conjugate of product
product of conjugates)
 conjugate of λ is a eigenvalue
 Conjugate of x is an eigenvector
Slide 5.3- 15© 2012 Pearson Education, Inc.

16. Complex Eigenvalues – Theorem 9
Let A be a real 2x2 matrix with complex
eigenvalue λ = a – bi (b ≠ 0) and an
associated eigenvector v in C2. Then
A = PCP-1 where
P = [Re(v) Im(v)]
𝐶 =
𝑎 −𝑏
𝑏 𝑎
Slide 5.3- 16© 2012 Pearson Education, Inc.