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coeffMatrices -- gets coefficient matrices for a matrix of linear forms

Description

Given a linear matrix pencil $M = I_d + x_1A_1 + ... + x_nA_n$, this method returns the list of matrices $A_1, ..., A_n$.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing
i2 : M = id_(R^3) + random(R^3,R^{3:-1})

o2 = | 9/2x+9/4y+3/4z+1 3/7x+6/7y+6z      10x+3/2y+7/8z    |
     | 7/4x+7/9y+7/10z  5/4x+2/9y+3/10z+1 5/6x+5y+2/5z     |
     | 7/10x+7/3y+7z    3/7x+5y+10/9z     5/3x+7/2y+2/5z+1 |

             3      3
o2 : Matrix R  <-- R
i3 : coeffs = coeffMatrices M

o3 = {{1} | 9/2  3/7 10  |, {1} | 9/4 6/7 3/2 |, {1} | 3/4  6    7/8 |}
      {1} | 7/4  5/4 5/6 |  {1} | 7/9 2/9 5   |  {1} | 7/10 3/10 2/5 |
      {1} | 7/10 3/7 5/3 |  {1} | 7/3 5   7/2 |  {1} | 7    10/9 2/5 |

o3 : List
i4 : M - sum(#gens R, i -> R_i*coeffs#i)

o4 = | 1 0 0 |
     | 0 1 0 |
     | 0 0 1 |

             3      3
o4 : Matrix R  <-- R

Caveat

This method does not return the constant term, or coefficients of terms of degree $> 1$.

Ways to use coeffMatrices:

  • coeffMatrices(Matrix)

For the programmer

The object coeffMatrices is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/DeterminantalRepresentations.m2:1184:0.