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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 x + y is irreducible over R, but can be written as

${\displaystyle \left\left=(x^{2}+y^{2})\left}$

References

1. Eisenbud, David (1980-01-01). "Homological algebra on a complete intersection, with an application to group representations". Transactions of the American Mathematical Society. 260 (1): 35. doi:10.1090/S0002-9947-1980-0570778-7. ISSN 0002-9947.
2. Crisler, David; Diveris, Kosmas, Matrix Factorizations of Sums of Squares Polynomials (PDF)

External links

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

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