How do you invert a 3D matrix? (Tensor inversion)

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To invert a 3D matrix (tensor) for solving the system of equations Ax=b, where A is an n x m x p tensor, one approach is to convert the tensor into a 2D matrix format by indexing it as [n x (m*p)]. This transformation allows the application of standard matrix inversion techniques. The discussion highlights the challenge of finding clear methods for tensor inversion, similar to regular matrix inversion. Additionally, the possibility of using least-squares methods is questioned but not fully explored. Understanding tensor manipulation is crucial for effectively solving these types of equations.
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I have three systems of equations in the form of Ax=b, where there are three different b-vectors, three different A-matrices, all of which use the same x-vector (A1x=b1, A2x=b2,A3x=b3). The goal is to solve for x. I can also write this as a tensor product: b_ij = sum_k (A_ijk x_k), where I would want to invert A to solve for X. I'm familar with regular linear systems where a is a 2D matrix and I could use a least-squares approach, MLEM, etc.
I would like to solve a system of systems of equations Ax=b where A is an n x m x p tensor (3D) matrix, x is a vector (n x 1), and b is a matrix (n x p). I haven't been able to find a clear walk-through of inverting a tensor like how one would invert a regular matrix to solve a system of linear equations. (or an iterative technique like MLEM).

Attached is a typed up version of the equations except with different variables where b = N, A = R, and x = S:

Figure.png
 
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Why can't one just apply least-squares?

Okay, convert ##r_{i,j,k}## to a ##[n\times m p ]## matrix. It's just indexing.
 
Whoops, I'll check this, thank you!
 

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