derivation of Sylvester’s matrix for the resultant

Sylvester’s matrix representationPlanetmathPlanetmath of the resultantMathworldPlanetmath can be easily derived by thinking of polynomialsMathworldPlanetmathPlanetmath as linear equations in powers of x. For ease of exposition, we shall consider the case where p is of second orderPlanetmathPlanetmath and q is of third order but, once the basic idea has been grasped, it is trivial to extend it to polynomials of any orders whatsoever.

Let us start with the polynomial equations p(x)=0 and q(x)=0. Written out in full, they look like


These two equations can be combined into a single matrix equation:


Since square matricesMathworldPlanetmath enjoy properties which matrices of arbitrary size do not, we will add more equations so as to come up with a new matrix equation involving a square matrices. There is no harm in adding an equation of the forn xkp(x)=0 or xkq(x)=0 to the system p(x)=0,q(x)=0 because the enlarged system will have exactly the same solutions as the original system of two equations. Consider the system


This system may be written as a matrix equation


Now we have a square matrix. One important property of matrix equations involving square matrices is that they only have non-trivial solutions when the determinantMathworldPlanetmath of the matrix vanishes. The system p(x)=0,q(x)=0 only has a solution when p and q have a common root. Hence the determinant will vanish whenever p and q have a common root.

Note that, at this stage, we cannot jump to the converseMathworldPlanetmath conclusionMathworldPlanetmath that p and q always have a common root when the determinant vanishes. All we can say is that, if the determinant vanishes, there will be some non-zero vector in the kernel of the matrix, but we cannot say that the vector will be of the special form (x4,x3,x2,x,1) that appears in the system. To assert the converse conclusion, we need to first prove that the determinant indeed equals the resultant. For this proof, please see the entry proof that Sylvester’s determinant equals the resultant.

Title derivationPlanetmathPlanetmath of Sylvester’s matrix for the resultant
Canonical name DerivationOfSylvestersMatrixForTheResultant
Date of creation 2013-03-22 14:36:47
Last modified on 2013-03-22 14:36:47
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 4
Author rspuzio (6075)
Entry type Derivation
Classification msc 13P10