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Positive Semidefinite matrix

Positive Semidefinite matrix. A is a positive semidefinite matrix. (also called nonnegative definite matrix). Positive definite matrix. A is a positive definite matrix. Negative semidefinite matrix. A is a negative semidefinite matrix. Negative definite matrix.

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Positive Semidefinite matrix

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  1. Positive Semidefinite matrix A is a positive semidefinite matrix (also called nonnegative definite matrix)

  2. Positive definite matrix A is a positive definite matrix

  3. Negative semidefinite matrix A is a negative semidefinite matrix

  4. Negative definite matrix A is a negative definite matrix

  5. Positive semidefinite matrix A is real symmetric matrix A is a positive semidefinite matrix

  6. Positive definite matrix A is real symmetric matrix A is a positive definite matrix

  7. Question Is It true that ? Yes

  8. Proof of Question ?

  9. Proof of Question ?

  10. Fact 1.1.6 • The eigenvalues of a Hermitian (resp. positive semidefinite , positive definite) matrix are all real (resp. nonnegative, positive)

  11. Proof of Fact 1.1.6

  12. Exercise From this exercise we can redefinite: H is a positive semidefinite

  13. 注意 A is symmetric

  14. 注意 之反例 is not symmetric

  15. Proof of Exercise

  16. Remark • Let A be an nxn real matrix. If λ is a real eigenvalue of A, then there must exist a corresponding real eigenvector. • However, if λ is a nonreal eigenvalue of A, then it cannot have a real eigenvector.

  17. Explain of Remark p.1 • A, λ : real Az= λz, 0≠z (A- λI)z=0 By Gauss method, we obtain that z is a real vector.

  18. Explain of Remark p.2 • A: real, λ is non-real Az= λz, 0≠z z is real, which is impossible

  19. Elementary symmetric function kth elementary symmetric function

  20. KxK Principal Minor kxk principal minor of A

  21. Lemma p.1

  22. Lemma p.2

  23. Explain Lemma

  24. The Sum of KxK Principal Minors

  25. Theorem

  26. Proof of Theorem p.1

  27. Proof of Theorem p.2

  28. Rank P.1 rankA:=the maximun number of linear independent column vectors =the dimension of the column space = the maximun number of linear independent row vectors =the dimension of the row space result result

  29. Rank P.2 rankA:=the number of nonzero rows in a row-echelon (or the reduced row echlon form of A)

  30. Rank P.3 rankA:=the size of its largest nonvanishing minor (not necessary a principal minor) =the order of its largest nonsigular submatrix. See next page

  31. Rank P.4 1x1 minor Not principal minor rankA=1

  32. Theorem Let A be an nxn sigular matrix. Let s be the algebraic multiple of eigenvalue 0 of A. Then A has at least one nonsingular (nonzero)principal submatrix(minor) of order n-s.

  33. Proof of Theorem p.1

  34. Geometric multiple Let A be a square matrix and λ be an eigenvalue of A, then the geometric multiple of λ=dimN(λI-A) the eigenspace of A corresponding to λ

  35. Diagonalizable

  36. Exercise A and have the same characteristic polynomial and moreover the geometric multiple and algebraic multiple are similarily invariants.

  37. Proof of Exercise p.1

  38. Proof of Exercise p.2 (2)Since A and have the same characteristic polynomial, they have the same eigenvalues and the algebraic multiple of each eigenvalue is the same.

  39. Proof of Exercise p.3

  40. Explain: geom.mult=alge.mult in diagonal matrix

  41. Fact For a diagonalizable(square) matrix, the algebraic multiple and the geometric multiple of each of its eigenvalues are equal.

  42. Corollary Let A be a diagonalizable(square) matrix and if r is the rank of A, then A has at least one nonsingular principal Submatrix of order r.

  43. Proof of Corollary p.1

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