Matrix factorization of a polynomial

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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.^[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

\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]

References

^ 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): 35–64. doi:10.1090/S0002-9947-1980-0570778-7. ISSN 0002-9947.
^ Crisler, David; Diveris, Kosmas (2016-10-21). "Matrix Factorizations of Sums of Squares Polynomials" (PDF). Northfield, Minnesota: St. Olaf College. Archived from the original (PDF) on 2020-07-06.

External links

A Mathematica implementation of an algorithm to matrix-factorize polynomials

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[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): 35–64. doi:10.1090/S0002-9947-1980-0570778-7. ISSN 0002-9947.

[2] Crisler, David; Diveris, Kosmas (2016-10-21). "Matrix Factorizations of Sums of Squares Polynomials" (PDF). Northfield, Minnesota: St. Olaf College. Archived from the original (PDF) on 2020-07-06.

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