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

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. the eigenspace of A corresponding to λ
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