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Chapter 6 Eigenvalues.

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Presentation on theme: "Chapter 6 Eigenvalues."— Presentation transcript:

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., )

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

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