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Chapter one

Chapter one. Matrix Theory Background. 1.Hermitian and real symmetric matrix. 1.Hermitian and real symmetric matrix. adjA. Symmetric and Hermitian matrices. Symmetric matrix: Hermitian matrix: :Complex symmetric matrix. Symmetric and Hermitian matrices.

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Chapter one

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  1. 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:1.1.1 p. 1

  21. Fact:1.1.1 p. 2

  22. Fact:1.1.1 p. 3

  23. Fact:1.1.1 p. 4 • Corresponding real version also hold.

  24. Fact:1.1.1 p. 5 • If , in addition ,m=n, then • Corresponding real version also hold.

  25. Fact:1.1.2 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:1.1.2 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. Proof of Theorem 1.1.4 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 1.1.4 p.3 • Partition B and C conformally with A as Then (*) implies that for any positive integer

  31. Proof of Theorem 1.1.4 p.4 Notice that It follows that M=0, since K is nonsingular Then A=BC is the same as

  32. Proof of Theorem 1.1.4 p.5 This yields K=NR, and hence N and R are nonsigular. Taking k=1 in (*), we have

  33. Proof of Theorem 1.1.4 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. Proof of Theorem 1.1.4 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. Proof of Theorem 1.1.4 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. Proof of Theorem 1.1.4 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 1.1.4 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.

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

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

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