In mathematics, a matrix factorization of a polynomial is a technique for factoring
irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
s with
matrices
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the ...
.
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now k ...
proved that every
multivariate real-valued polynomial ''p'' without linear terms can be written as ''AB'' = ''pI'', where ''A'' and ''B'' are
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
and ''I'' is the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
. Given the polynomial ''p'', the matrices ''A'' and ''B'' can be found by elementary methods.
Example
The polynomial ''x''
2 + ''y''
2 is irreducible over R
'x'',''y'' but can be written as
: