Let a non null real matrix of dimension . Let the ideal generated by the polynomials . Show that this polynomials are a Gröbner base of the ideal I for some lexicographic order.
Consider the lexicographic order .
Reduce the matrix in row echelon form using row operations; The polynomial set produces the same ideal, and cannot be reduced any further.
Thus, this set (and equivalently the starting set) is a Groebner basis.