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Outline Hermitian Matrix Unitary Matrix Toeplitz Matrix

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Presentation on theme: "Outline Hermitian Matrix Unitary Matrix Toeplitz Matrix"— Presentation transcript:

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


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