Difficult linear algebra problem

[Klandhee]

Hi, my problem is simple enough to write down but (to me) seems quite difficult to solve.

My equation is as follows

A[x1 x2] = I.

Here I is some known matrix, and A is an operator which applies a shifting matrix and sums. That is A[x1 x2] = s1x1 + s2x2, where s1 and s2 are two shifting matrices (continuously it can be thought of as convolving with a delta function). x1 and x2 are two unknown matrices of the same dimension as I. Ultimately I wish to find a matrix form for A so that I can invert it and obtain x1 and x2

So as you can see [x1 x2] can be thought of as a "stack" of matrices, or a 3D matrix (or a tensor?). However I'm very unfamiliar with the mathematics of tensors so one idea I had was to convert x1 and x2 into columns (i.e., just shopping the matrix into slices and adding one ontop of the other). That way [x1 x2] would be a matrix, and I would have lost no information.

From here, however, I am very confused and not sure where to go.

If anyone has any ideas on what to do (or if this problem is impossible) it would be GREATLY appreciated, thanks!

 

[rasmhop]

This doesn't sound like homework so I will assume it's not (and therefore not feel bad about providing a "solution").

I'm not sure exactly what you mean by a shifting matrix. The only definition I know of is matrices which are 0 everywhere except on precisely one diagonal either below or above the main diagonal where they are 1. For instance in the 2x2 case
\left[\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right], \quad\left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]
are the shift matrices and in the 3x3 case we have:
\left[\begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right], \quad\left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right]
If this is the case, then we can easily see that it is impossible to solve in general. For example define:
A[x_1,x_2] = \left[\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right] x_1 +\left[\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right]x_2
I = \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right]
Then we have infinitely many solutions of the form
x_1 = \left[\begin{array}{cc} a & b \\ c & d \end{array} \right] \qquad x_2 = \left[\begin{array}{cc} -a & -b \\ e & f \end{array} \right]
for arbitrary reals a,b,c,d,e,f.