Can someone explain / link me to a linear algebra worked problem where I can see how these work. I've searched and given their statistics and matrix specialty uses, can't find any ready examples.
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3It's best to give a specific context (such as a reference or link), because symbols can be used for more than one thing, and you don't explicitly say that you want say the Kronecker product or the Hermitian matrix (as Antoni -- likely correctly -- assumed) – Glen_b Jan 26 '16 at 01:31
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$\bigotimes$ is the Kronecker product of two matrices. In this post I used the Kronecker product to understand the model matrices in mixed effects models.
$A^* = A^H = \bar A^T$ is the Hermitian or conjugate transpose of a matrix $A$.
A good example of the use of Hermitian matrices is in obtaining the inverse of a Fourier matrix in the Fast Fourier Transform (FFT). For an $n \times n$ Fourier matrix, $F_n$, the inverse is going to be the Hermitian conjugate, such that $\frac{1}{n}\,F_n^H\,F_n = I.$
For example, a $4\times4$ Fourier matrix ($n=4$) would be:
$$\large \begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & W & W^2 & W^3\\ 1 & W^2 & W^4 & W^6 \\ 1 & W^3 & W^6 & W^9 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & e^{i\,\frac{2\pi}{4}} & e^{i\,2\frac{2\pi}{4}} & e^{i\,3\frac{2\pi}{4}}\\ 1 & e^{i\,2\frac{2\pi}{4}} & e^{i\,4\frac{2\pi}{4}} & e^{i\,6\frac{2\pi}{4}} \\ 1 & e^{i\,3\frac{2\pi}{4}} & e^{i\,6\frac{2\pi}{4}} & e^{i\,9\frac{2\pi}{4}} \end{bmatrix}= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & i & -1 & -i\\ 1 & -1 & 1 & -1 \\ 1 & -i & -1 & i \end{bmatrix} $$
And the Hermitian of this matrix:
$$F_4=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & i & -1 & -i\\ 1 & -1 & 1 & -1 \\ 1 & -i & -1 & i \end{bmatrix}\implies F_4^H = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i\\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{bmatrix} $$
Antoni Parellada
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