Matrix factorization of a polynomial

In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as AB = pI, where A and B are square matrices and I is the identity matrix. Given the polynomial p, the matrices A and B can be found by elementary methods.

Example

The polynomial x² + y² is irreducible over R[x,y], but can be written as

$\left[\begin{array}{cc}x& -y\\ y& x\end{array}\right]\left[\begin{array}{cc}x& y\\ -y& x\end{array}\right]=(x^2+y^2)\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

External links

• A Mathematica implementation of an algorithm to matrix-factorize polynomials