Matrix factorization of a polynomial

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Short description: Mathematical technique

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.[2]

Example

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

[x−yyx][xy−yx]=(x2+y2)[1001]

References

  1. ↑ Eisenbud, David (1980-01-01). "Homological algebra on a complete intersection, with an application to group representations" (in en). Transactions of the American Mathematical Society 260 (1): 35–64. doi:10.1090/S0002-9947-1980-0570778-7. ISSN 0002-9947. https://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1980-0570778-7. 
  2. ↑ Crisler, David; Diveris, Kosmas (2016-10-21). "Matrix Factorizations of Sums of Squares Polynomials" (PDF). St. Olaf College. https://pages.stolaf.edu/wp-content/uploads/sites/46/2017/01/MFE1.pdf. 




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