1. Chapter 6 Eigenvalues

2. Outlines System of linear ODE (Omit) Diagonalization
Hermitian matrices
Ouadratic form
Positive definite matrices

3. Motivations
To simplify a linear dynamics such that it is as simple as possible.
To realize a linear system characteristics
e.g.,
the behavior of system dynamics.

4. §6-1 Eignvalues & Eignvectors

5. Example: In a town, each year
30% married women get divorced
20% single women get married
In the 1st year, 8000 married women 2000 single women.
Total population remains constant
, where
represent the numbers of married & single
women after i years, respectively.

6. If
Question: Why does converge?
Why does it converge to the same limit
vector even when the initial condition is
different?

7. Ans: Choose a basis
Given an initial for some
for example,
Question: How does one know choosing such a basis?

8. Def: Let . A scalar is said to be an
eigenvalue or characteristic value of A if
such that
The vector is said to be an eigenvector
or characteristic vector belonging to .
is called an eigen pair of A.
Question: Given A, How to compute eigenvalues
& eigenvectors?

9. characteristic polynomial of A, of degree n in
is an eigen pair of
Note that, is a polynomial, called
characteristic polynomial of A, of degree n in
Thus, by FTA, A has exactly n eigenvalues including
multiplicities.
is a eigenvector associated with eigenvalue while is eigenspace of A.

10. Example:
To find the eigenspace of 2:( i.e., )

11. To find the eigenspace of 3(i.e., )
Let

12. Let , then

13. Let
If is an eigenvalue of A with eigenvector
Then
This means that is also an eigen-pair of A.

14. Let
where are eigenvalues of A.
(i) Let
(ii) Compare with the coefficient of
, we have

15. Theorem 6.1.1: Let A & B be n×n matrices, if B is similar to A, then and consequently A & B have the same eigenvalues.
Pf: Let for some nonsingular matrix S.

16. §6-3 Diagonalization

17. Diagonalization
Goal: Given find a nonsingular matrix
S, such that is a diagonal
matrix.
Question1: Are all matrices diagonalizable?
Question2: What kinds of A are diagonalizable?
Question3: How to find S if A is diagonalizable?

18. NOT all matrices are diagonalizable
e.g., Let
If A is diagonalizable,
nonsingular matrix S,

19. To answer Q2,
Suppose that A is diagonalizable.
nonsingular matrix S,
Let
This gives a condition for diagonalizability and a way to
find S.

20. Theorem6.3.1: If are distinct eigenvalues of an
matrix A with corresponding eigenvectors ,
then are linearly independent.
Pf: Suppose that are linearly dependent
not all zero,
Suppose that

21. Theorem 6.3.2: Let is diagonalizable
A has n linearly independent eigenvectors.
Note : Similarity transformation
Change of coordinate
diagonalization

22. Remarks: Let , and
(i) is an eigenpair of A for
(ii) The diagonalizing matrix S is not unique because
Its columns can be reordered or multiplied by
an nonzero scalar
(iii) If A has n distinct eigenvalues , A is diagonalizable.
If the eigenvalues are not distinct , then may or may
not diagonalizable depending on whether A has n
linearly independent eigenvectors or not.
(iv)

23. Example: Let
For
Let

24. Def:
If an matrix A has fewer than n linearly
independent eigenvectors,we say that A is defective.
e.g.
(i) is defective
(ii) is defective

25. Example 4: Let
A & B both have the same eigenvalues
Nullity (A-2I)=1
The eigenspace associated with has only one dimension.
A is NOT diagonalizable
However,
Nullity (B-2I)=2
B is diagonalizable

26. Question:
Are the following matrices diagonalizable ?

27. The Exponential of a Matrix
Motivation:The general solution of is
The unique solution of
Question: What is and how to compute ?

28. Note that
Define

29. Suppose that A is diagonalizable with

30. Example 6: Compute
Sol: The eigenvalues of A are
with eigenvectors

31. §6-4 Hermitian Matrices

32. Hermitian matrices
Let , then A can be written as
, where
e.g. ,

33. Let , then
e.g. , 

34. Def:
(a) A is said to be Hermitian if
(b) A is said to be skew-Hermitian if
(c) A is said to be unitary if
( i.e. its column vectors form an orthonormal set in )

35. (ii) Let and be two eigenpairs of A,
Theorem6.4.1: Let , then
(i)
(ii) eigenvectors belonging to distinct eigenvalues
are orthogonal.
Pf：(i) Let be an eigenpair of A,
(ii) Let and be two eigenpairs of A,

36. (ii) Let and be two eigenpairs of A,
Theorem6.4.1
Pf：
(ii) Let and be two eigenpairs of A,

37. Theorem: Let and
then
Pf：(i) Let be an eigenpair of A,

38. Theorem6.4.3 (Schur’s Theorem): Let ,
then unitary matrix , is upper triangular.
Pf：
The proof is by mathematical induction on n.
(i) The result is obvious if n=1;
(ii) Assume the hypothesis holds for k×k matrices;
(iii) let A be a (k+1)×(k+1) matrix.

39. Proof of Schur’s Theorem
Let be an eigenpair of A with
Using the Gram-Schmidt process, construct an orthonormal
basis of
Let

40. Proof of Schur’s Theorem
By the induction hypothesis (ii)

41. Theorem6.4.4: (Spectral Theorem)
If , then unitary matrix U that diagonalizes A .
Pf：By Theorem ,
unitary matrix ,
where T is upper triangular .

42. Cor.6.4.5: Let A be real symmetric matrix .
Then (i)
(ii) an orthogonal matrix U,
is a diagonal matrix.
proof：

43. Example 4:
Find an orthogonal matrix U that diagonalizes A.
Sol : (i)
(ii)

44. Example 4:
Sol :
(iii) By Gram-Schmidt process,

45. Note：If A has orthonormal eigenbasis
Question:In addition to Hermitian matrices ,
is there any other matrices possessing
orthonormal eigenbasis?
Note：If A has orthonormal eigenbasis
where U is unitary &diagonal.

46. Def: A is said to be normal if
Remark：
Hermitian, Skew- Hermitian and Unitary matrices are all normal.

47. then ,where U is unitary & diagonal.
Theorem6.4.6: A is normal A possesses an
orthonormal eigenbasis
Pf：
If A has an orthonormal eigenbasis,
then ,where U is unitary & diagonal.

48. proof of Theorem6.4.6:
By Th.6.4.3, unitary U,
Compare the diagonal elements of