In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring.
The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain).
Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus: The Sylvester matrix associated to p and q is then the
matrix constructed as follows: Thus, if m = 4 and n = 3, the matrix is: If one of the degrees is zero (that is, the corresponding polynomial is a nonzero constant polynomial), then there are zero rows consisting of coefficients of the other polynomial, and the Sylvester matrix is a diagonal matrix of dimension the degree of the non-constant polynomial, with the all diagonal coefficients equal to the constant polynomial.
In a paper of 1853, Sylvester introduced the following matrix, which is, up to a permutation of the rows, the Sylvester matrix of p and q, which are both considered as having degree max(m, n).
These matrices are used in commutative algebra, e.g. to test if two polynomials have a (non-constant) common factor.
In such a case, the determinant of the associated Sylvester matrix (which is called the resultant of the two polynomials) equals zero.
The solutions of the simultaneous linear equations where
, comprise the coefficient vectors of those and only those pairs
, respectively) which fulfill where polynomial multiplication and addition is used.
This means the kernel of the transposed Sylvester matrix gives all solutions of the Bézout equation where
Consequently the rank of the Sylvester matrix determines the degree of the greatest common divisor of p and q: Moreover, the coefficients of this greatest common divisor may be expressed as determinants of submatrices of the Sylvester matrix (see Subresultant).