Exact deconvolution of two matrices using numerical techniques Suppose that I am given two $n \times m$ matrices $\bf{A}$ and $\bf{C}$, and let $\bf{B}$ be a matrix that is convolved with $\bf{A}$, such that: $\bf{A} * B = C$ In the above, $*$ is the convolution operator. I am searching for a numerical procedure (and ideally a reference book or paper on the mathematics of the operation) that will allow me to determine $\bf{B}$, given $\bf{A}$ and $\bf{C}$. I know both $\bf{A}$ and $\bf{C}$ exactly. The $\bf{A}$ is the system input, whereas the $\bf{C}$ is the system output. Although the convolution is a product in the Fourier domain, I might be able to determine $\bf{B}$ by division. However, using the 2D Fast Fourier Transform (FFT), do I have to pad the matrices in the spatial domain or deal with division by zero? How do I do this in an elegent manner? Is there a way to do this in Matlab using the `fft2` and `ifft2` functions?

In your case, there is no noise in the output, then the Fourier method is a safe way for deconvolution: first zero-pad $\mathbf{A}$ to the size of $\mathbf{C}$ (without zero-padding will leads to result corresponding to circulant convolution); then compute the spectra using `fft2`; divide the spectrum of $\mathbf{C}$ by the one of $\mathbf{A}$, then you can get the deconvolution result by using `ifft2`.

However, if the output contains noise, then the Fourier method may not a proper way for deconvolution, since division in frequency domain could amplify the noise and contaminate the results. The better way is, as Seyhmus Güngören mentioned, to convert the deconvolution problem into a system of linear equations or optimization problem with certain regularizations, then use iterative algorithms to solve the equations or optimization problem.