derivation of Sylvester’s matrix for the resultant
Sylvester’s matrix representation of the resultant can be easily derived by thinking of polynomials as linear equations in powers of . For ease of exposition, we shall consider the case where is of second order and 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 and . Written out in full, they look like
Since square matrices 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 or to the system 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 determinant of the matrix vanishes. The system only has a solution when and have a common root. Hence the determinant will vanish whenever and have a common root.
Note that, at this stage, we cannot jump to the converse conclusion that and 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 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||derivation of Sylvester’s matrix for the resultant|
|Date of creation||2013-03-22 14:36:47|
|Last modified on||2013-03-22 14:36:47|
|Last modified by||rspuzio (6075)|