ideals to and from matrices

ideals to and from matrices

Sections:

forming an ideal from a matrix

forming a matrix from an ideal

forming an ideal from a matrix

After defining a matrix (see input a matrix) , M, the ideal generated by the entries of the matrix is obtained using the command ideal.

i1 : R = ZZ/101[a..e];

i2 : M = matrix{{a^2*b-c^2, a*b^2-d^3, c^5-d},{a^2*b, b*c*d, c^5}}



o2 = | a2b-c2 ab2-d3 c5-d |

     | a2b    bcd    c5   |



             2       3

o2 : Matrix R  <--- R

i3 : ideal M



             2     2   2      2    3          5       5

o3 = ideal (a b - c , a b, a*b  - d , b*c*d, c  - d, c )



o3 : Ideal of R

forming a matrix from an ideal

In much the same way we can construct a 1 by n (where n is the number of generators of I), matrix from the n generators of an ideal I using the command, generators.

i4 : I = ideal(a^2*b-c^2+c*d, a*b^2-b*d^3, c^5,d+e);



o4 : Ideal of R

i5 : generators I



o5 = | a2b-c2+cd -bd3+ab2 c5 d+e |



             1       4

o5 : Matrix R  <--- R

The abbreviation gens can be used for generators.