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Understanding Hermitian and Symmetric Matrices: Theoretical Foundations and Key Concepts

This chapter delves into the properties of Hermitian and real symmetric matrices, including their definitions, forms, and significant characteristics. It explores skew-symmetric and skew-Hermitian matrices, highlighting crucial relationships between these concepts. Further, it discusses eigenvalues within linear transformations and presents essential theorems that establish connections between Hermitian matrices and other matrix types. Examples and proofs are provided to illustrate these ideas, emphasizing the implications for real vector spaces and transformations.

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Understanding Hermitian and Symmetric Matrices: Theoretical Foundations and Key Concepts

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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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