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.[1] Given the polynomial p, the matrices A and B can be found by elementary methods.
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]=(x2+y2) \left[\begin{array}{cc} 1&0\\ 0&1\end{array}\right]