1. Spectrum of A σ(A)=the spectrum of A =the set of all eigenvalues of A

2. Proposition

3. Similarity matrix If,then we say that A is transformed to B under similarity via similarity matrix P

4. Exercise 1.2.4 If are similar over C, then A and B are similar over R. 組合矩陣理論 第一 章 Exercise.doc

5. Proof of Exercise 1.2.4

6. Schur’s unitary triangularilation Theorem unitarily similar can be in any prescribed order

7. Normal matrix e.g Hermitian matrix, real symmetric matrix, unitary matrix, real orthogonal matrix, skew- Hermitian matrix, skew-symmetric matrix. 強調與 complex symmetric matric 作區 別

8. Remark about normal matrix Normal matices can not form a subspace.

9. Fact (*) for Normal matrix Proof in next page

11. Spectrum Thm for normal matix 注意 Appling Schur’s unitary triangulariation Theorem to prove.

12. Real Version of Spectrum Thm for normal matix It is normal. The proof is in next page

14. Proposition for eigenvalue

15. Proof of privious Proposition

16. 1.3 Jordan Form and Minimal Polynomial

17. Elementary Jordan Block main diagonal elementary jordan block super diagonal sub diagonal

18. It is Nilpotent matrix.(see next page)

20. Jordan Matrix jordan matrix

21. Jordan Canonical Form Theorem unique up to the ordering of elementary Jordan blocks along the block diagonal. A is similar to a jordan matrix If A is real with only real eigenvalues, then the similarity matrix can be taken to be real By Exercise 1.2.4

22. Observation 1 for Jordan matrix the jordan matrix of A

23. Observation 2 for Jordan matrix the proof in next page

27. Observation 3 for Jordan matrix

28. Observation 4 for Jordan matrix

29. Observation 5 for Jordan matrix Given counter example in next page The algebraic and geometric multiple of λ can not determine completely the Jordan structure corresponding to λ

30. Assume that 1 is an eigenvalue of A and geometric multiple of 1 is 3 algebraic multiple of 1 is 5 then 3 blocks in corresponding to λ the sum of sizes of these blocks is 5 Therefore (see next page)

31. or

32. Annihilating polynomial for A In next page we show that A has an annihilating polynomial. Let p(t) be a polynomial. If p(A)=0, then we say p(t) annihilates A and p(t) is an annihilating polynomial for A

34. Minimal polynomial of A The minimal polynomial of A is monic polynomial of least degree that annihilates A and is denoted by the proof in next page

36. Caley-Hamilton Theorem This Theorem implies that

37. Minimal Polynomial when A~B the proof in next page

39. Mimimal poly. of Jordan matrix Given example to explain in next page

42. Similarly,

44. Mimimal poly. of Jordan matrix Proof in next page

49. index of eigenvalue p.1 See next page

50. index of eigenvalue p.2

51. index of eigenvalue p.3

52. Observation 6 for Jordan matrix p.1 ….

53. Observation 6 for Jordan matrix p.2 the proof in next page

55. Observation 7 for Jordan matrix p.2 the proof in next page The number of blocks in of size ≧ k is

58. See next page 組合矩陣理 論 第一章 Exercise.doc

59. Jordan structures for The Jordan structure of A corresponding to and that corresponding to are the same. Because

60. The proof is in next page.

62. Permutation similarity The proof is in next page

65. similarly Prove d in next page

67. similarly

68. Theorem 1.3.4

70. By Exercise 1.2.4 組合矩陣理論 第 一 章 Exercise.doc