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

## Synopsis

• Usage:
coeffMatrices(M)
• Inputs:
• M, , of linear forms
• Outputs:
• a list, of matrices

## 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+1/2y+9/4z+1 7/9x+7/10y+1/2z 2/3x+y+2z | | 1/2x+y+3/4z 7/10x+7/3y+7z+1 6x+5/4y+2/9z | | 3/2x+3/4y+7/4z 3/7x+5/2y+6/7z 5x+3/10y+z+1 | 3 3 o2 : Matrix R <--- R i3 : coeffs = coeffMatrices M o3 = {{1} | 9/2 7/9 2/3 |, {1} | 1/2 7/10 1 |, {1} | 9/4 1/2 2 |} {1} | 1/2 7/10 6 | {1} | 1 7/3 5/4 | {1} | 3/4 7 2/9 | {1} | 3/2 3/7 5 | {1} | 3/4 5/2 3/10 | {1} | 7/4 6/7 1 | 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 .