Matrix Theory Background Chapter one Matrix Theory Background

1.Hermitian and real symmetric matrix

1.Hermitian and real symmetric matrix

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

Symmetric and Hermitian matrices

Hermitian matrices

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

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

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

Skew-Hermitian matrices

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

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

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.

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

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

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

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

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

Fact:1.1.1 p. 1

Fact:1.1.1 p. 2

Fact:1.1.1 p. 3

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

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

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.

Fact:1.1.2 p. 1 Therefore, AB is Hermitian if and only if AB=BA

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

(*) 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 (*)

Note that J is nilpotent and K is invertible. 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.

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

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

This yields K=NR, and hence N and R 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

It follows that A is similar to 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

a product of two Hermitian matrice 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

Theorem 3.13 says that if A is similar 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

Where J(λ) is the Jorden block with 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:

Proof of Theorem 1.1.4 p.10 which is equal to a product of two Hermitian matrices

=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.

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

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

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

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