Sylvester Matrix

For two polynomials P_1(x)=a_mx^m+...+a_0 and P_2=b_nx^n+...+b_0 of degrees m and n, respectively, the Sylvester matrix is an (m+n)×(m+n) matrix formed by filling the matrix beginning with the upper left corner with the coefficients of P_1(x), then shifting down one row and one column to the right and filling in the coefficients starting there until they hit the right side. The process is then repeated for the coefficients of P_2(x).

The Sylvester matrix can be implemented in the Wolfram Language as:

  SylvesterMatrix1[poly1_, poly2_,  var_] :=
    Function[{coeffs1, coeffs2}, With[
      {l1 = Length[coeffs1], l2 = Length[coeffs2]},
          NestList[RotateRight, PadRight[coeffs1,
            l1 + l2 -  2], l2 - 2],
          NestList[RotateRight, PadRight[coeffs2,
            l1 + l2 - 2], l1 - 2]
      Reverse[CoefficientList[poly1, var]],
      Reverse[CoefficientList[poly2, var]]

For example, the Sylvester matrix for P_1(x)=a_3x^3+a_2x^2+a_1x+a_0 and P_2(x)=b_2x^2+b_1x+b_0 is

 [a_3 a_2 a_1 a_0 0; 0 a_3 a_2 a_1 a_0; b_2 b_1 b_0 0 0; 0 b_2 b_1 b_0 0; 0 0 b_2 b_1 b_0].

The determinant of the Sylvester matrix of two polynomials is the resultant of the polynomials.

SylvesterMatrix is an (undocumented) method for the Resultant function in the Wolfram Language (although it is documented in Trott 2006, p. 29).

See also

Determinant, Resultant

Explore with Wolfram|Alpha


Akritas, A. G. "Sylvester's Forgotten Form of the Resultant." Fib. Quart. 31, 325-332, 1993.Akritas, A. G. "Sylvester's Form of the Resultant and the Matrix-Triangularization Subresultant prs Method." Proceedings of the Conference on Computer Aided Proofs in Analysis, Cincinnati, Ohio, March, 1989 (Ed. K. R. Meyer and D. S. Schmidt.) IMA Volumes in Mathematics and its Applications, 28, 5-11, 1991.Laidacker, M. A. "Another Theorem Relating Sylvester's Matrix and the Greatest Common Divisor." Math. Mag. 42, 126-128, 1969.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, p. 28, 2006.

Referenced on Wolfram|Alpha

Sylvester Matrix

Cite this as:

Weisstein, Eric W. "Sylvester Matrix." From MathWorld--A Wolfram Web Resource.

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