getActionMatrix -- computes or retrieves a template's action matrix

Usage:

A = getActionMatrix E
Inputs:
- E, an instance of the type EliminationTemplate, an elimination template
Optional inputs:
- AdjustParams (missing documentation) => ..., default value true,
- MonomialOrder (missing documentation) => ..., default value null,
- Strategy (missing documentation) => ..., default value null,
Outputs:
- A, a matrix, the action matrix

Description

For a zero-dimensional polynomial ideal $I \subset R$, multiplication by a general linear form $f \in R$ induces a linear map $R/I \to R/I$, which can be represented using an action matrix. For an ideal represented by a template matrix, the action matrix may be obtained from the template matrix by various triangularization schemes (RREF, LU Decomposition, etc.) The eigenvalues can be used to determine the (closed) points in the vanishing locus of $I$, as they give the values of $f$ on these points. In this implementation, the action matrix is cached inside the template to facilitate quicker computation.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x + y - 1, x^2 + y^2 - 1)

                        2    2
o2 = ideal (x + y - 1, x  + y  - 1)

o2 : Ideal of R

i3 : E = eliminationTemplate(x,I)

o3 =  action variable: x

o3 : EliminationTemplate

i4 : A = getActionMatrix E

o4 = | 0  0 |
     | -1 1 |

              2       2
o4 : Matrix QQ  <-- QQ

i5 : eigenvalues A

o5 = {1}
     {0}

o5 : VerticalList

Ways to use getActionMatrix:

getActionMatrix(EliminationTemplate)

For the programmer

The object getActionMatrix is a method function with options.

The source of this document is in EliminationTemplates.m2:1019:0.

getActionMatrix -- computes or retrieves a template's action matrix

Description

See also

Ways to use getActionMatrix:

For the programmer