Is there an equivalent to the row-echelon transform (and by extension, the reduced row-echelon transform) for a "3D matrix", or more generally an "n-dimensional matrix"? I want to say "Rank 3 tensor" but from what little I know about them I know they're not merely just "matrices but generalized to n-dimensions" so I feel like using that terminology might not be correct, but neither is "3D matrix", and I don't know of any other mathematical structure that gets the same point across. There's also "3D / n-dimensional array" but that term I've only seen used in programming (in this context).
For example, take a fully 3D volumetric version of the game "Lights Out". In the traditional 2D game the objective is to turn off all of the tiles on a 5x5 board; selecting a square not only toggles the one you picked but also the adjacent tiles. The same principle applies to the 3D version, except the tiles are cubes in a 5x5x5 "board", and adjacent cubes are toggled taking all three dimensions into account. (This is not to be confused with the "Lights Out Cube" which still uses 2D tiles, just mapped onto the surface of a cube) The 2D version can be solved mathematically using Gauss-Jordan elimination to get reduced-row echelon form, and I wonder if a fully volumetric 3D version can be solved with some similar method, and what it'd look like.
