Gaussian elimination
Generalizations

Gaussian elimination can be performed over any field, not just the real numbers.

Buchberger's algorithm is a generalization of Gaussian elimination to systems of polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination, manifesting as the choice to work from left to right when selecting pivot positions.

Computing the rank of a tensor of order greater than 2 is NP-hard. Therefore, if P ≠ NP, there cannot be a polynomial time analog of Gaussian elimination for higher-order tensors (matrices are array representations of order-2 tensors).
detectable_content:number_placeholders[4]