1. 4. The Eigenvalue

2. Overview
As we shall see, the eigenvalue problem is of great practical importance in mathematics and applications. In section 4.1 we introduce the eigenvalue problem for the special case of (2×2) matrices; this special case can be handled using ideas developed in Chapter 1. In section 4.4 we move on to the general case, the eigenvalue problem for (n×n) matrices. The general case requires several results form determinant theory, and these are summarized in section 4.2. If you are familiar with these results, you can proceed directly to the general (n×n) case in section 4.4.

3. Core sections The eigenvalue problem for (2×2) matrices
Determinants and the eigenvalue problem
Eigenvalues and the characteristic polynomial
Eigenvectors and eigenspaces
Complex eigenvalues and eigenvectors
Similarity transformations and diagonalization

4. 4.1 The eigenvalue problem for (2×2) matrices
The eigenvalue problem, the topic of this chapter, is a problem of considerable theoretical interest and wide-ranging application.
Solving systems of differential equations
Analyzing population growth models
Calculating powers of matrices
Diagonalizing linear transformation
Simplifying and describing the graphs of quadratic forms in to and three variables

5. 4.1 The eigenvalue problem for (2×2) matrices
Definition4.1.1: For an (n×n) matrix A, find all scalars λsuch that the equaion
Ax=λx
has a nonzero solution, such a scalar λis called an eigenvalue of A, and any nonzero (n×1) vector x satisfying Ax=λx is called an eigenvector corresponding to λ.

6. The Geometric interpretation of Eigenvalue and eigenvector

7. The calculation of Eigenvalue and eigenvector
The eigenvalue problem consists of two parts:

8. Eigenvalue and eigenvectors for (2×2) matrices
B is singular if and only if ru-st=0

9. Example: Find all eigenvalues and eigenvectors of A, where
Exercise P

10. 4.2 Determinants and the eigenvalue problem
Determinants of (2×2) Matrics
The determinants of A, denoted by det(A), is the number
vertical bar

11. 4.2 Determinants and the eigenvalue problem
Determinants of (3×3) Matrics
The determinants of A, denoted by det(A), is the number
The determinants of a 3×3 matrix is defined to be the weighted sum of three 2×2 determinants. Similarly, the determinant of n×n matrix will be defined as the weighted sum of n determinants each of order [(n-1) ×(n-1)]

12. 4.2 Determinants and the eigenvalue problem
Determinants of (3×3) Matrics

13. 4.2 Determinants and the eigenvalue problem
Minors and Cofactors (子式，代数余子式)
Definition 3 Let A=aij be an (n×n) matrix. The [(n-1) ×(n-1)] matrix that results form removing the rth row and sth column from A is called a minors of A and is designated by Mrs

14. 4.2 Determinants and the eigenvalue problem
Determinants of (3×3) Matrics
The determinants of A, denoted by det(A), is the number

15. 4.2 Determinants and the eigenvalue problem
Determinants of (3×3) Matrics
The determinants of A, denoted by det(A), is the number
余子式
cofactors (or signed minors) 代数余子式

16. 4.2 Determinants and the eigenvalue problem
Determinants of (3×3) Matrics
The determinants of A, denoted by det(A), is the number
cofactors expansion corresponding to the first row

17. 4.2 Determinants and the eigenvalue problem
Second-column cofactor expansions

18. 4.2 Determinants and the eigenvalue problem
Determinants of (n×n) Matrics
the determinants of A, denoted by det(A), is the number
Cofactor expansion

19. 4.2 Determinants and the eigenvalue problem
Determinants of (n×n) Matrics

20. 4.2 Determinants and the eigenvalue problem
Determinants of (n×n) Matrics

21. 4.2 Determinants and the eigenvalue problem
Determinants and singular Matrics
A is singular if and only if det(A) =0

22. 4.3 Elementary operations and determinants

23. 4.4 Eigenvalues and the characteristic polynomial
Find all scalarsλsuch that |A-λI|=0 (or |λI - A |=0 ). (Such scalars are the eigenvalues of A, characteristic equation )
Given an eigenvalue λ, find all nonzero vectors x such that (A- λI)x=0. (Such vectors are the eigenvectors corresponding to the eigenvalue λ.)
Example: Use the singularity test given in Eq.(1) to determine the eigenvalues of the matrix A, where

24. The characteristic polynomial 特征多项式
Theorem: Let A be an (n×n) matrix. Then det(A-λI) is a polynomial of degree n in λ.
Definition: Let A be an (n×n) matrix. The nth-degree polynomial, p(λ) =det(A-λI) is is called the characteristic polynomial for A.
Theorem11: Let A be an (n×n) matrix, and letλbe an eigenvalue of A. Then
λk is an eigenvalue of Ak;
If A is nonsingular, then 1/ λ is an eigenvalue of A-1;
If c is any scalar, then λ+c is an eigenvalue of A+cI.

25. Example Solution eigenvector eigenvalue Suppose that
Determine the eigenvalue of A
Solution
is eigenvalue of A
eigenvector
eigenvalue

26. Theorem12: Let A be an (n×n) matrix
Theorem12: Let A be an (n×n) matrix. Then A and have the same eigenvalues.
Theorem13: Let A be an (n×n) matrix. Then A is singular if and only if λ=0 is an eigenvalue of A.
Note: if A is singular, then the eigenvectors corresponding to λ=0 are in the null space of A.

27. Exercise P ,13

29. 4.5 Eigenvectors and Eigenspaces
As we saw in the previous section, we can find the eigenvalues of a matrix A by solving the characteristic equation (A-tI)=0. Once we know the eigenvalues, the familiar technique of Gaussian elimination can be employed to find the eigenvectors that correspond to the various eigenvalues.
In particular, the eigenvectors corresponding to an eigenvalue
of A are the nonzero solutions of

30. Eigenvector is nonzero vector
4.5 Eigenvectors and Eigenspaces
Eigenvector is nonzero vector

31. Eigenspaces and Geometric Multiplicity
4.5 Eigenvectors and Eigenspaces
Eigenspaces and Geometric Multiplicity
The relationship between Geometric Multiplicity and algebraic Multiplicity

32. 4.5 Eigenvectors and Eigenspaces
The relationship between Geometric Multiplicity and algebraic Multiplicity

33. 4.5 Eigenvectors and Eigenspaces
Defective matrices
For application it will be important to know whether an (n×n) matrix A has a set of n linearly independent eigenvetors. As we will see later, if A is an (n×n) matrix and if some eigenvalue of A has a geometric multiplicity that is less than its algebraic multiplicity, then A will not have a set of n linearly independent eigenvectors. Such a matrix is called defective

34. 4.5 Eigenvectors and Eigenspaces
Defective matrices
Definition7: Let A be an (n×n) matrix. If there is an eigemvalue λ of A such that the geometric multiplicity of λ is less than the algebraic multiplicity of λ, the A is called a defective matrix
EXAMPLE5: find all the eigenvalues and eigenvectors of the matrix A. Also, determine the algebraic and geometric multiplicities of the eigenvalues.

35. 4.5 Eigenvectors and Eigenspaces
Defective matrices

36. Theorem15: Let u1,u2,…,uk be eigenvectors of an (n×n) matrix A corresponding to distinct eigenvalues λ1, λ2, …, λk that is,
Auj=λjuj for j=1,2, …,k; k≤n
λi≠λj for i ≠ j; 1 ≤i,j ≤k.
Then {u1,u2,…,uk } is a linearly independent set.
Corollary: Let A be an (n×n) matrix. If A has n distinct eigenvalues, then A has a set of n linearly independent eigenvectors.

37. Proof: since u1 ≠0,then set{u1} is trivially linearly independent, if {u1,u2,…,uk } were linearly dependent, then there would exist an integer m, 2 ≤m≤k. such that

39. Hence we have contradicted the assumption that there is an m, m≤k , such that S2 is linearly dependent. Thus {u1,u2,…,uk } is linearly independent

40. 4.7 Similarity transformations and diagonalization
In Chapter 1, we saw that two linear systems of equations have the same solution if their augmented matrices are row equivalent. In this chapter, we are interested in identifying classes of matrices that have the same eigenvalues.
If an (n×n) matrices A and B have the same characteristic polynomial, then A and B have the same eigenvalues.

41. This show that the similar matrices have the same characteristic

42. 4.7 Similarity transformations and diagonalization
Definition: The (n×n) matrices A and B are said to be similar (denoted A ～ B) if there is a nonsingular (n×n) matrix S such that B=S-1AS.

43. This show that the similar matrices have the same characteristic polynomial

44. 4.7 Similarity transformations and diagonalization
Theorem18: If A and B are similar (n×n) matrices, then A and B have the same eigenvalues. Moreover, these eigenvalues have the same algebraic multiplicity.
the similar matrices have the same characteristic polynomial, it is not true that two matrices with the same characteristic polynomial are necessarily similar

45. 4.7 Similarity transformations and diagonalization
the similar matrices have the same eigenvalue, they do not generally have the same eigenvectors

46. Identity matrix I only similar to I
matrix aI only similar to aI

47. Similarity is an equivalence relation on the space of square matrices.
Similar matrices share many properties:
rank
determinant
trace
eigenvalues (though the eigenvectors will in general be different)
characteristic polynomial

48. In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,

49. where aij represents the entry on the ith row and jth column of A
where aij represents the entry on the ith row and jth column of A. Equivalently, the trace of a matrix is the sum of its eigenvalues, making it an invariant with respect to a change of basis. This characterization can be used to define the trace for a linear operator in general. Note that the trace is only defined for a square matrix (i.e. n×n).

51. Diagonalization
Let A be an (n×n) matrix, the computation involving A can be simplified if we know that A is similar to a diagonal matrix.

52. Diagonalization

53. Diagonalization
Theorem19: An (n×n) matrix A is diagonalizable if and only if A possesses a set of n linearly independent eigenvectors.
Theorem20: Let A be an (n×n) matrix with n distinct eigenvalues. Then A is diagonalizable.

58. Example 1
Example 2 Calculate

59. Diagonalization
Theorem19: An (n×n) matrix A is diagonalizable if and only if A possesses a set of n linearly independent eigenvectors.
Theorem20: Let A be an (n×n) matrix with n distinct eigenvalues. Then A is diagonalizable.

60. Example 4

61. Example

66. Orthogonal matrices(正交矩阵)
A remarkable and useful fact about symmetric matrices is that they are always diagonalizable. Moreover, the diagonalization of a symmetric matrix A can be accomplished with a special type of matrix know as an orthogonal matrix.
Definition 9

68. Example 5 Verify the matrices are orthogonal

69. Definition: A real (n×n) matrix Q is called an orthogonal matrix if Q is invertible and Q-1=QT.
Theorem: Let Q be an (n×n) orthogonal matrix.
If x is in Rn, then ||Qx||=||x||.
If x and y are in Rn , then (Qx)T(Qy)=xTy.
Det(Q)=±1.
Diagonalizaiton of symmetric matrices
We conclude this section by showing that every symmetric matrix can be diagonalized by an orthogonal matrix.

70. Diagonalizaiton of symmetric matrices
We conclude this section by showing that every symmetric matrix can be diagonalized by an orthogonal matrix. Several approaches can be used to establish this diagonalization result. We choose to demonstrate it by first stating a special case of a theorem known as Schur’s theorem

71. Theorem: Let A be an (n×n) real matrix.
If A is symmetric, then there is an orthogonal matrix Q such that QTAQ=D, where D is diagonal.
If QTAQ=D, where Q is orthogonal and D is diagonal, then A is symmetric matrix.
Corollary: Let A be a real (n×n) symmetric matrix. It is possible to choose eigenvectors u1,u2,…,un for A such that {u1,u2,…,un} is an orthonormal basis for Rn.

72. EXAMPLE Diagonalization

73. Characteristic polynomial
engenvalues

74. To
engenvector

77. are different
orthogonal
Unitisation

78. suppose
then