Fundamental of algebra mathematics course

1. Chapter 4
Fundamentals of Matrix Algebra By Gregory Hartman
Do we ever have the case where where a is some scalar?
Central Question

2. 4.1 Eigenvalues and
Eigenvectors

3. Predefinition Workup
Page 3

4. Page 4

5. More Math
Page 5
where 𝐵 = 𝐴 − 𝜆𝐼
Recall 𝐵𝑥 = 0 from 2.4 not
Back then 𝑥 = 0
Now we’re looking for
an eigenvector which
by definition cannot
be 0.
Thus we want a solutions to 𝐴 − 𝜆𝐼 𝑥 = 0 other than 𝑥 = 0
Recall:
Theorem 8: any invertible matrix − 𝐴 − 𝜆𝐼 − only has one solution.
Therefore, 𝐴 − 𝜆𝐼 needs to not be invertible.
Theorem 16: noninvertible matrices all have a determinant of 0.
Thus det 𝐴 − 𝜆𝐼 needs to equal 0.
Now we move forward by finding 𝜆 such that det 𝐴 − 𝜆𝐼 = 0
and

6. What are eigenvalues?
• Given a matrix, A, x is the eigenvector and  is the
corresponding eigenvalue if Ax = x
• A must be square and the determinant of A -  I must be
equal to zero
Ax - x = 0 iff (A - I) x = 0
• Trivial solution is if x = 0
• The nontrivial solution occurs when det(A - I) = 0
• Are eigenvectors unique?
• If x is an eigenvector, then x is also an eigenvector and  is
an eigenvalue
A(x) = (Ax) = (x) = (x)

7. Example 84
Page 7
Solution
Problem

8. Example 85
Page 8
Problem
Solution

9. Calculating the Eigenvectors/values
• Expand the det(A - I) = 0 for a 2 x 2 matrix
• For a 2 x 2 matrix, this is a simple quadratic equation with two solutions
(maybe complex)
• This “characteristic equation” can be used to solve for x
 
  
    0
0
0
det
0
1
0
0
1
det
det
21
12
22
11
22
11
2
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4 a
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10. Eigenvalue example
• Consider,
• The corresponding eigenvectors can be computed as
• For  = 0, one possible solution is x = (2, -1)
• For  = 5, one possible solution is x = (1, 2)
   
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5
,
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)
4
1
(
0
2
2
4
1
)
4
1
(
0
4
2
2
1
2
2
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12
22
11
22
11
2


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5
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1
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0
0
4
2
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1
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


11. Definition
Page 11

12. Key Idea
Page 12

13. Put Another Way

14. Example 86
Page 14
Fine the eigenvalues of A, and for each eigenvalue, find an eigenvector where
Solution
Problem

15. Example 86
Page 15
Fine the eigenvalues of A, and for each eigenvalue, find an eigenvector where
Solution
Problem

16. Example 86
Page 16
Fine the eigenvalues of A, and for each eigenvalue, find an eigenvector where
Solution
Problem
Summary

17. Example 86
Page 17
Fine the eigenvalues of A, and for each eigenvalue, find an eigenvector where
Solution
Problem

19. 4.2 Properties of Eigenvalues
and Eigenvectors
Page 19
Note regarding section material: matrices with a trace of 0 are important.

20. Example 90
Page 20
Solution
Problem

21. Page 21
The Eigenvectors and Eigenvalues of A and A-1
If 𝜆 is an eigenvalue of A, then 1/𝜆 is an eigenvalue of A-1.
The corresponding eigenvectors are the same.

22. Page 22
The Eigenvectors and Eigenvalues of A and AT
The Eigenvalues of A and AT are the same, but eigenvectors are different.

23. Page 23

24. Page 24
Example 92
Problem
Solution

25. Page 25
Theorem

26. Observe that a 2x2 matrix may have
only one eigenvalue.
Eigenvalues can also be complex numbers.
These facts should not be surprising,
since eigenvalues are solutions of
polynomials.

27. Page 27
Finding Eigenvalues in Real Applications
The eigenvalues of a matrix A can be determined by finding the roots of the characteristic polynomial. Explicit
algebraic formulas for the roots of a polynomial exist only if the degree n is 4 or less. Therefore, for matrices of
order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula and must
be computed by approximate numerical methods.
We would expect that for larger matrices a computer would be used to factor the characteristic polynomials. In
reality, this method is unreliable as roundoff errors cause unpredictable results. Furthermore, before computing
the characteristic polynomial, we need to compute the determinant, also a very expensive process. Efficient,
accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the
advent of the QR algorithm in 1961.
Question: how then are eigenvalues found?
Answer: Via iterative processes that transform matrix A into a matrix that is almost an upper triangular
matrix. The more iterations, the better approximation.
Once the value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero
solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients.

28. Page 28
Example Compute the eigenvalues and eigenvectors for matrix A. How about A2 , A-1 , and A + 4I ?
𝐴 =
2 −1
−1 2