1. Matrix Theory Background
Chapter one
Matrix Theory Background

2. 1.Hermitian and real symmetric matrix

3. 1.Hermitian and real symmetric matrix

4. adjA

5. Symmetric and Hermitian matrices
Symmetric matrix:
Hermitian matrix:
:Complex symmetric matrix

6. Symmetric and Hermitian matrices

7. Hermitian matrices

8. Hermitian matrices Form
Let H be a Hermitian matrix, then H
is the following form conjugate compelx number

9. Skew-Symmetric and Skew- Hermitian
Skew-symmetric matrix:
Skew-Hermitian matrix:

10. Skew-symmetric matrices Form
Let A be a skew-symmetric matrix,
then A is the following form r

11. Skew-Hermitian matrices

12. Skew-Hermitian matrices Form
Let H be a skew-Hermitian matrix,
then H is the following form

13. Symmetric and Hermitian matrices
If A is a real matrix, then
For real matrice,
Hermitian matrices and (real) symmetric
matrices are the same.

14. Symmetric and Hermitian matrices
Since every real Hermitian matrix is real
symmetric, almost every result for Hermitian
matrices has a corresponding result for
real symmetric matrices.

15. Given Example for almost p.1
A result for Hermitian matrice:
If A is a Hermitian matrix, then there is a
unitary matrix U such that
We must by a parallel proof obtain the
following result for real symmetric matrices

16. Given Example for almost p.2
A result for real symmetric matrice:
If A is a real symmetric matrix,
then there is a real orthogonal matrix P
such that

17. Given another Example for almost
A result for complex matrice:
If A is a complex matrix, such that
A counterexample for real matric:

18. Eigenvalue of a Linear Transformation p.1
Eigenvalues of a linear transformation on a
real vector space are real numbers. This
is by definition.

19. Eigenvalue of a Linear Transformation p.2
We can extend T as following:
Similarly, we can extend A as following

20. Fact: p. 1

21. Fact: p. 2

22. Fact: p. 3

23. Fact: p. 4
Corresponding real version also hold.

24. Fact: p. 5
If , in addition ,m=n, then
Corresponding real version also hold.

25. Fact: p. 1
If A is Hermitian, then is Hermitian
for k=1,2,…,n
If A is Hermitian and A is nonsingular, then
is Hermitian.

26. Fact: p. 1
Therefore, AB is Hermitian if and only if AB=BA

27. Theorem 1.1.4
A square matrix A is a product of two
Hermitian matrices if and only if
A is similar to

28. (*) Proof of Theorem 1.1.4 p.1 Necessity: Let A=BC,
where B and C are Hermitian matrices
Then
and inductively for any positive integer k
(*)

29. Note that J is nilpotent and K is invertible.
Proof of Theorem p.2
We may write, without loss of generality
visa similarity
where J and K contain Jordan blocks of
eigenvalues 0 and nonzero, respectively.
Note that J is nilpotent and K is invertible.

30. Proof of Theorem p.3
Partition B and C conformally with A as
Then (*) implies that for any positive integer

31. Proof of Theorem p.4
Notice that
It follows that M=0, since K is nonsingular
Then A=BC is the same as

32. This yields K=NR, and hence N and R
Proof of Theorem p.5
This yields K=NR, and hence N and R
are nonsigular.
Taking k=1 in (*), we have

33. It follows that A is similar to
Proof of Theorem p.6
which gives
or, since N is invertible,
In other words, K is similar to
Since J is similar to ,
It follows that A is similar to

34. a product of two Hermitian matrice
Proof of Theorem p.7
Sufficiency:
Notice that
This says that if A is similar to a product
of two Hermitian matrices, then A is in fact
a product of two Hermitian matrice

35. Theorem 3.13 says that if A is similar
Proof of Theorem p.8
Theorem 3.13 says that if A is similar
to that , then the Jorden blocks of
nonreal eigenvalues of A occur in cojugate
pairs. Thus it is sufficient to show that

36. Where J(λ) is the Jorden block with
Proof of Theorem p.9
Where J(λ) is the Jorden block with
λ on the diagonal, is similar to a product
of two Hermitian matrices.
This is seen as follows:

37. Proof of Theorem p.10
which is equal to a product of two
Hermitian matrices

38. =the set of all nxn Hermitian matrices
=the set of all nxn skew Hermitian
matrix.
This means that every skew Hermitian
matrix can be written in the form iA
where A is Hermitian and conversely.

39. Given a skew Hermitian matrix B,
B=i(-iB)
where -iB is a Hermitian matrix.

40. ( also ) form a real vector space
under matrix addition and multiplication
by real scalar with dimension.

42. H(A)= :Hermitian part of A
S(A)= :skew-Hermitian part of A

43. Re(A)= :real part of A
Im(A)= :image part of A