The SylvesterMatrix(P, Q) function constructs the Sylvester Matrix of polynomials P and Q in the same indeterminate.

The SylvesterMatrix(P, Q, x) function is used when P and Q are multivariate polynomials; the main indeterminate of P and Q can be specified by using the optional parameter x.

If S := SylvesterMatrix(P, Q, x) and n is the sum of the degrees of polynomials P and Q, then S is an n x n Matrix with $S_{j,j+i}=\mathrm{coeff}\left(P\,x\,\mathrm{deg}\left(P\right)-i\right)$ when i = 0..deg(P), j = 1..deg(Q) and $S_{\mathrm{deg}\left(Q\right)+j,j+i}=\mathrm{coeff}\left(Q\,x\,\mathrm{deg}\left(Q\right)-i\right)$ when i = 0..deg(Q), j = 1..deg(P).

The determinant of the Sylvester Matrix is equal to resultant(P, Q, x).

If the compact option (cpt) is included in the calling sequence as just the symbol compact or in the form compact=true, then the result is built by using a shape function designed to minimize storage.  If the option is omitted or entered as compact=false, a full rectangular Matrix is constructed.  Generally, if space is not a consideration, the full rectangular form (the default) is more efficient.

Note:  If the compact form is selected, any datatype specification in the constructor options is silently ignored.  The datatype of any data retrieved from the constructed Matrix is determined by the data used to build the Matrix.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

This function is part of the LinearAlgebra package, and so it can be used in the form SylvesterMatrix(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[SylvesterMatrix](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$p≔a+bx\:$

$q≔c+dx+e{x}^2\:$

$A≔\mathrm{SylvesterMatrix}\left(p\,q\,x\right)$

$A≔\left[\begin{array}{ccc}b& a& 0\\ 0& b& a\\ e& d& c\end{array}\right]$

$\mathrm{Determinant}\left(A\right)$

$e{a}^2-abd+b^{2}c$

$\mathrm{resultant}\left(p\,q\,x\right)$

$e{a}^2-abd+b^{2}c$

$\mathrm{SylvesterMatrix}\left(3-x+x^2\,2x\right)$

$\left[\begin{array}{ccc}1& \mathrm{-1}& 3\\ 2& 0& 0\\ 0& 2& 0\end{array}\right]$