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In mathematics, a '''matrix factorization of a polynomial''' is an technique for factoring [[irreducible polynomial]]s with [[Matrix (mathematics)|matrices]]. [[David Eisenbud]] proved that every [[Multivariate polynomial|multivariate real-valued polynomial]] ''p'' without linear terms can be written as a ''AB'' = ''pI'', where ''A'' and ''B'' are [[square matrices]] and ''I'' is the [[identity matrix]].<ref>{{Cite journal|last=Eisenbud|first=David|date=1980-01-01|title=Homological algebra on a complete intersection, with an application to group representations|url=https://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1980-0570778-7|journal=Transactions of the American Mathematical Society|language=en|volume=260|issue=1|pages=35|doi=10.1090/S0002-9947-1980-0570778-7|issn=0002-9947|doi-access=free}}</ref> Given the polynomial ''p'', the matrices ''A'' and ''B'' can be found by elementary methods.<ref>{{Citation|last1=Crisler|first1=David|title=Matrix Factorizations of Sums of Squares Polynomials|url=https://pages.stolaf.edu/wp-content/uploads/sites/46/2017/01/MFE1.pdf|last2=Diveris|first2=Kosmas}}</ref>
In mathematics, a '''matrix factorization of a polynomial''' is a technique for factoring [[irreducible polynomial]]s with [[Matrix (mathematics)|matrices]]. [[David Eisenbud]] proved that every [[Multivariate polynomial|multivariate real-valued polynomial]] ''p'' without linear terms can be written as a ''AB'' = ''pI'', where ''A'' and ''B'' are [[square matrices]] and ''I'' is the [[identity matrix]].<ref>{{Cite journal|last=Eisenbud|first=David|date=1980-01-01|title=Homological algebra on a complete intersection, with an application to group representations|url=https://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1980-0570778-7|journal=Transactions of the American Mathematical Society|language=en|volume=260|issue=1|pages=35|doi=10.1090/S0002-9947-1980-0570778-7|issn=0002-9947|doi-access=free}}</ref> Given the polynomial ''p'', the matrices ''A'' and ''B'' can be found by elementary methods.<ref>{{Citation|last1=Crisler|first1=David|title=Matrix Factorizations of Sums of Squares Polynomials|url=https://pages.stolaf.edu/wp-content/uploads/sites/46/2017/01/MFE1.pdf|last2=Diveris|first2=Kosmas}}</ref>


*Example:
*Example:

Revision as of 21:05, 21 October 2022

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 a 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

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)