Sylvester matrix

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.
Definition.
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 formula_2 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. If "m" = "n" = 0, then the Sylvester matrix is the empty matrix with zero rows and zero columns.
A variant.
The above defined Sylvester matrix appears in a Sylvester paper of 1840. 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").
This is thus a formula_6-matrix containing formula_7 pairs of rows. Assuming formula_8 it is obtained as follows:
Thus, if "m" = 4 and "n" = 3, the matrix is:
The determinant of the 1853 matrix is, up to sign, the product of the determinant of the Sylvester matrix (which is called the resultant of "p" and "q") by formula_12 (still supposing formula_13).
Applications.
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 converse is also true.
The solutions of the simultaneous linear equations
where formula_15 is a vector of size formula_16 and formula_17 has size formula_18, comprise the coefficient vectors of those and only those pairs formula_19 of polynomials (of degrees formula_20 and formula_21, 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 formula_23 and formula_24.
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).