# Which one can be accompanied by linear filters?

I have a matrix :

$$\begin{bmatrix} 1 & 2 & 3\\ 1 & 4 & 5\\ 2 & 6 & 7 \end{bmatrix}$$

After doing operation 1 , I get

$$\begin{bmatrix} 0 & 0 & 0\\ 1 & 2 & 3\\ 1 & 4 & 5 \end{bmatrix}$$

However operation 2 on same matrix will give : $$\begin{bmatrix} 0 & 1 & 2\\ 0 & 3 & 4\\ 1 & 5 & 6 \end{bmatrix}$$

Clearly, operation 2 reduces each matrix value by 1, while operation 1 shifts one row down.

Now I see that operation 1 is simply subtraction and thus can be applied using linear filter. But I am not using to linear filters on matrix. Can anyone guide me through this?

TL;DR: Time shift in doable with LTI systems. Subtracting $$1$$ is non linear.

In discrete systems, integer shifts can be implemented with filters that are zero everywhere, except at the shift location. This can be implemented via circulant matrices, akin to convolution. This can be made compatible with Operation 1 (taking border effects into accounts).

The second operation adds one to every value. This is not linear (but affine), and therefore does not match with linear filters.