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Spectrum of A

Spectrum of A. σ(A)=the spectrum of A =the set of all eigenvalues of A. Proposition. Similarity matrix. If ,then we say that A is transformed to B under similarity via similarity matrix P. Exercise 1.2.4. If are similar over C,

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Spectrum of A

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  1. Spectrum of A σ(A)=thespectrum 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 can be in any prescribed order unitarily similar

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

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

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

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

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

  12. Proposition for eigenvalue

  13. Proof of privious Proposition

  14. 1.3 Jordan Form and Minimal Polynomial

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

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

  17. Jordan Matrix jordan matrix

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

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

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

  21. Observation 3 for Jordan matrix

  22. Observation 4 for Jordan matrix

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

  24. 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)

  25. or

  26. Annihilating polynomial for A 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 In next page we show that A has an annihilating polynomial.

  27. 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

  28. Caley-Hamilton Theorem This Theorem implies that

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

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

  31. Similarly,

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

  33. index of eigenvalue p.1 See next page

  34. index of eigenvalue p.2

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