<s>[INST] What does $\bigotimes$ and $X^*$ mean? 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.  
 [/INST] $\bigotimes$ 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:
$$F_4=\begin{bmatrix} 
1 & 1 & 1 & 1 \\
1 & i & -1 & -i\\
1 & -1 & 1 & -1 \\
1 & -i & -1 & i 
\end{bmatrix}\implies F_4^H =
\begin{bmatrix} 
1 & 1 & 1 & 1 \\
1 & -i & -1 & i\\
1 & -1 & 1 & -1 \\
1 & i & -1 & -i 
\end{bmatrix}
$$
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