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It is not clear what are you asking but I will try answer both things. Deriving the Matrix Inversion Lemma The Matrix Inversion Lemma goes as: $${\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} V {A}^{-1}$$ Deriving it is by utilizing these useful identities: \begin{align} U + U C V {A}^{-1} U &... 2 The matrices \hat{\mathbf{M}} and \tilde{\mathbf{M}} are constructed in such a way that the relation \mathbf{M}\mathbf{x}=\mathbf{y} implies \hat{\mathbf{M}}\hat{\mathbf{x}}=\hat{\mathbf{y}} and \tilde{\mathbf{M}}\tilde{\mathbf{x}}=\tilde{\mathbf{y}}. Consequently, for constructing the matrix \tilde{\mathbf{M}}, each element m_{kl} of \... 1 Marcus' answer is perfect! For any wireless channel, y = Hx + n, where: y = received signal x = transmitted signal n = noise in the channel For an AWGN channel, output 'y' = input 'x' plus 'noise' (AWGN) i.e. y = x + n Therefore, for AWGN channel, channel matrix or H-matrix is the identity matrix. 1 The assertion "almost always seem to be expressed in integers?" does not seem to be true, in my opinion. However, such kernels are pretty frequent in codes. They are quantized both in support (limited discrete support) and amplitude (signed integer values). 3\times 3 masks with integer coefficients, as you showed, are very familiar, albethey ... 1 The Laplacian kernel with the 4 in the middle results from summing second derivatives along the two axes ([1,-2,1]). Those are the right values to use, you can show this by writing out the math for the second derivative and set the distance h to 1 (or search for discrete approximation to derivative). This kernel hasn’t been rounded, the values just happen to ... 1 If  H  is a matrix form of Circular Convolution then it is a Circulant Matrix. Being a Circulant Matrix means it can be diagonalized by the Fourier Matrix  {F} : H = {F}^{H} D F  Where the matrix $D$ id a Diagonal Matrix with the Fourier Coefficients of $\mathcal{F} \left( h \right)$ on it main diagonal. Also pay attention that we use the ...
Have you taken a look at the documentation - section 2.2.4 discusses a Linear Kalman filter model that is very similar to the one you described. From that example you see that: The resulting $\bf{A}$ matrix does not depend on the process noise The $\bf{A}$ matrix only depends on the size of the time step. The $\bf{Q}$ matrix only depends on the size of the ...