In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. 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). Sylvester matrices are named after James Joseph Sylvester who first used them in 1853.
Definition Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus:
p ( z ) = p 0 + p 1 z + p 2 z 2 + ⋯ + p m z m , q ( z ) = q 0 + q 1 z + q 2 z 2 + ⋯ + q n z n . {\displaystyle p(z)=p_{0}+p_{1}z+p_{2}z^{2}+\cdots +p_{m}z^{m},\;q(z)=q_{0}+q_{1}z+q_{2}z^{2}+\cdots +q_{n}z^{n}.}
The Sylvester matrix associated to p and q is then the ( n + m ) × ( n + m ) {\displaystyle (n+m)\times (n+m)} matrix constructed as follows:
if n > 0, the first row is:
( p m p m − 1 ⋯ p 1 p 0 0 ⋯ 0 ) . {\displaystyle {\begin{pmatrix}p_{m}&p_{m-1}&\cdots &p_{1}&p_{0}&0&\cdots &0\end{pmatrix}}.}
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, shifting the coefficients one column to the right each time and setting the other entries in the row to be 0. if m > 0 the (n + 1)th row is:
( q n q n − 1 ⋯ q 1 q 0 0 ⋯ 0 ) . {\displaystyle {\begin{pmatrix}q_{n}&q_{n-1}&\cdots &q_{1}&q_{0}&0&\cdots &0\end{pmatrix}}.}
the following rows are obtained the same way as before. Thus, if m = 4 and n = 3, the matrix is:
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