# matrixFactorizationK -- Knoerrer matrix factorization from a bilinear form X*transpose Y

## Synopsis

• Usage:
(M1, M2) = matrixFactorizationK(X,Y)
• Inputs:
• X, , row matrix of linear forms with constant coefficients
• Y, , row matrix of linear forms with linear coefficients of same length as X
• Outputs:
• M1, ,
• M2, ,

## Description

Produces a matrix factorization (M1,M2) of the bilinear form X*transpose Y. It does this by specializing the formula given by Knoerrer for $\sum X_i*Y_i$.

 i1 : kk=ZZ/101 o1 = kk o1 : QuotientRing i2 : n=2 o2 = 2 i3 : R=kk[a_0..a_(binomial(n+2,2))] o3 = R o3 : PolynomialRing i4 : S=kk[x_0..x_(n-1),a_0..a_(binomial(n+2,2))] o4 = S o4 : PolynomialRing i5 : M=genericSymmetricMatrix(S,a_0,n) o5 = | a_0 a_1 | | a_1 a_2 | 2 2 o5 : Matrix S <--- S i6 : X=(vars S)_{0..n-1} o6 = | x_0 x_1 | 1 2 o6 : Matrix S <--- S i7 : Y=X*M o7 = | x_0a_0+x_1a_1 x_0a_1+x_1a_2 | 1 2 o7 : Matrix S <--- S i8 : (M1,M2)=matrixFactorizationK(X,Y) o8 = ({1} | -x_0 -x_1 |, {2} | -x_0a_0-x_1a_1 x_1 |) {0} | x_0a_1+x_1a_2 -x_0a_0-x_1a_1 | {2} | -x_0a_1-x_1a_2 -x_0 | o8 : Sequence i9 : M12=M1*M2 o9 = {1} | x_0^2a_0+2x_0x_1a_1+x_1^2a_2 0 | {0} | 0 x_0^2a_0+2x_0x_1a_1+x_1^2a_2 | 2 2 o9 : Matrix S <--- S

## Ways to use matrixFactorizationK :

• "matrixFactorizationK(Matrix,Matrix)"

## For the programmer

The object matrixFactorizationK is .