1. Corollary 1.2.9 If A is diagonalizable and rank A=r,
then A has at least one rxr nonsingular
principal submatrix.

2. permutation matrix

3. Proof

4. Proof

5. Proof

6. Fact 1.2.10 p.1 If A is mxn matrix and r is the size of
the largest nonsingular submatrix.Then
rank A=r
If B is a rxr nonsingular submatrix, then there are permutation matrices

7. Fact p.2
P,Q such that
(iii) If , in addition, m=n and B is principal then
may choose

8. Proof of Fact p.1

9. Proof of Fact p.2
kth row

10. Proof of Fact p.3

11. Proof of Fact p.4

12. Proof of Fact p.5

13. Theorem 1.2.13 If m=n ,and at least one of A or B is nonsigular,then
AB and BA are similar

14. Proof of Theorem p.1

15. Proof of Theorem p.2

16. Corollary 1.4.3 If A is a real symmetric matrix of rank r
then there is a permutation P and
rxr nonsingular principal submatrix M s.t

17. Proof of Corollary p.1

18. Proof of Corollary p.2

19. Usual Inner Product of

20. Unitary
U is said to be unitary if exists
and equals
i.e

21. Fact is unitary if and only if the columns of U form an
orthonormal basis of
proof: (see next page)

23. Real Orthogonal is real orthogonal if
i.e A real orthogonal matrix is a real
matrix which is unitary

24. Fact is real orthogonormal if and only if the columns of U form an
orthonormal basis of
proof: (see next page)

26. Fact A is unitarily diagonalized has a orthonormal basis
consisting of eigenvectors of A
proof: (in next page)

28. Fact A is diagonalizable has a basis consisting of eigenvectors of A
proof: (in next page)

30. Theorem 1.4.1(The spectral Thm for Hermitian matrices) p.1
?(未證)

31. Theorem 1.4.1(The spectral Thm for Hermitian matrices) p.2
If A is real sysmmetric, then
U can be chosen to be
real orthogonal matrix