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 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
- [math]\displaystyle{ \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] }[/math]
References
- ↑ 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. doi:10.1090/S0002-9947-1980-0570778-7. ISSN 0002-9947. https://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1980-0570778-7.
- ↑ Crisler, David; Diveris, Kosmas, Matrix Factorizations of Sums of Squares Polynomials, https://pages.stolaf.edu/wp-content/uploads/sites/46/2017/01/MFE1.pdf
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