- Sylvester matrix
mathematics, a Sylvester matrix is a matrix associated to two polynomials that gives us some information about those polynomials. It is named for James Joseph Sylvester.
Formally, let "p" and "q" be two polynomials, respectively of degree "m" and "n". Thus::The Sylvester matrix associated to "p" and "q" is then the matrix obtained as follows:
* the first row is::
* the second row is the first row, shifted one column to the right; the first element of the row is zero.
* the following (n-2) rows are obtained the same way, still filling the first column with a zero.
* the (n+1)-th row is::
* the following rows are obtained the same way as before.
Thus, if we put "m"=4 and "n"=3, the matrix is::
Those matrices are used in
commutative algebra, e.g. to test if two polynomials have a (non constant) common factor. Indeed, in such a case, the determinantof the associated Sylvester matrix (which is named the resultantof the two polynomials) equals zero. The converse is also true.
The solution of the simultaneous linear equations:where is a vector of size and has size , comprises the coefficient vectors of those and only those pairs of polynomials (of degrees and , respectively) which fulfill:(where polynomial multiplication and addition is used in this last line).This means the kernel of the transposed Sylvester matrix gives all solutions of the Bézout equation where and .
Consequently the rank of the Sylvester matrix determines the degree of the
greatest common divisorof and .:.
* [http://aix1.uottawa.ca/~jkhoury/elimination.htm Additional overview]
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