1. Outline Hermitian Matrix Unitary Matrix Toeplitz Matrix
Circulant Matrix

2. Hermitian Matrix Hermitian Matrix: A = AH Why define Hermitian matrix
Real function: f(x) = xHAx
Variance matrix: E[xxH]
Examples:
A + AH, AAH, AHA for any matrix A
Variance matrix, E[xxH]
Other examples of Hermitian matrix?

3. Hermitian Matrix Properties
A is Hermitian, if and only if xHAx real for all complex vector x
xHAx = (xHAx)* = xHAHx for all x
then A=AH
A and B are Hermitian, αA + βB is Hermitian for real α and β
prove by definition

4. Positive Definite Matrix
Definition of Hermitian matrix A
Function xHAx > 0, for all nonzero x
All eigenvalues of A are positive
A = UHΛU, and thus xHAx = (Ux)H Λ(Ux)
A = RHR for some n*n non-singular matrix R
we have that: R = Λ1/2U
PHAP is positive definite for all n*n non-singular matrix P: xH(PHAP)x = (Px)HA(Px)

5. Unitary Matrix All the columns and rows form orthogonal basis
It can be proved via definition
Linear independent rows/columns
We have that det(A) ≠ 0
U is unitary, if and only if the length of ||Ux|| is the same as the length of ||x||
||Ux||2 = xHUHUx = xHx = ||x||2

6. Unitary Matrix |det(U)| = 1, |eig(U)| = 1
det(UUH) = det(U) det(U)* = |det(U)|2
Ux = λx  ||x|| = ||Ux|| = ||λx|| = |λ|*||x||
A and B are unitary matrix, then A B is unitary
We have the following

7. Toeplitz Matrix Toeplitz Matrix
Matrix Representation for Moving Average Model
Hermitian Toeplitz Matrix: A = AH

8. Toeplitz Matrix Examples on the Toeplitz Matrix
Inter-symbol interference channel
Other examples on the Toeplize Matrix?

9. Circulant Matrix Circulant Matrix Representation
rows/columns are circulant
shift of other rows/columns
Eigenvalue Decomposition
for Circulant Matrix
FFT for the sequence
Eigenvectors, Fourier matrix

10. Circulant Matrix Eigenvalue Decomposition for Circulant Matrix
Eigenvalues: FFT of sequence

11. Circulant Matrix
Eigenvalue decomposition
Fourier transform

12. Circulant Matrix and Toeplitz Matrix
Circulant Matrix Approximation for Toeplitz Matrix
Frequency-domain Equalization
Eigenvalue decomposition for the circulant matrix

13. Circulant Matrix and Toeplitz Matrix
Circulant Matrix Approximation for Toeplitz Matrix
Frequency-domain Equalization
Eigenvalue decomposition for the circulant matrix