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LinearAlgebra

 BezoutMatrix
 construct the Bezout Matrix of two polynomials

 Calling Sequence BezoutMatrix(P, Q, x, options)

Parameters

 P, Q - polynomials in x x - (optional) name; the variable in polynomials P and Q options - (optional); constructor options for the result object method=symmetric - (optional) use the symmetric form of the Bezout matrix methodoptions=increasing_degree - (optional) order the Bezout matrix entries so that the null space is of the form 1, x, x^2, ... (the default is decreasing_degree)

Description

 • The BezoutMatrix(P, Q) command constructs the Bezout Matrix of the univariate polynomials P and Q in the same indeterminate.
 • The BezoutMatrix(P, Q, x) command 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.
 • Let $P=\sum _{i=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{p}_{i}{x}^{i}$ and $Q=\sum _{j=0}^{m}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{q}_{j}{x}^{j}$ where $m\le n$.  Then $B=\mathrm{BezoutMatrix}\left(P,Q,x\right)$ has entries

 ${B}_{m-i+1,j}=\mathrm{coeff}\left(\left(\sum _{r=i}^{m}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{q}_{m-r}{x}^{n-r}\right)\left(\sum _{r=1}^{i}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{p}_{n-r+1}{x}^{i-r}\right)-\left(\sum _{r=i}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{p}_{n-r}{x}^{n-r}\right)\left(\sum _{r=1}^{i}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{q}_{m-r+1}{x}^{i-r}\right),x,n-j\right)$ for $i=1..m$ and $j=1..n$ ${B}_{i,i-m-1+j}={q}_{m-j-1}$ for $i=m+1..n$ and $j=1..m+1$ ${B}_{i,j}=0$ for all other values of $i$ and $j$

 • If the option method=symmetric is chosen, then the matrix B is returned such that $\frac{P\left(x\right)Q\left(\mathrm{\eta }\right)-P\left(\mathrm{\eta }\right)Q\left(x\right)}{x-\mathrm{\eta }}=⟨\mathrm{...},{\mathrm{\eta }}^{2},\mathrm{\eta },1⟩·B·{⟨\mathrm{...},{x}^{2},x,1⟩}^{\mathrm{%T}}$.
 • If the option methodoptions=increasing_degree is chosen, the ordering of the vectors in this quadratic form are reversed.
 • If degree(Q) > degree(P), then BezoutMatrix(Q, P, x) is constructed.
 • The determinant of the Bezout Matrix is equal to $\mathrm{resultant}\left(P,Q,x\right)$.
 • If the option method=symmetric is chosen, then the determinant of the Bezout Matrix is a multiple of $\mathrm{resultant}\left(P,Q,x\right)$.
 • The constructor option 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).

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $p≔a+bx+c{x}^{2}$
 ${p}{≔}{c}{}{{x}}^{{2}}{+}{b}{}{x}{+}{a}$ (1)
 > $q≔d+ex+f{x}^{3}$
 ${q}{≔}{f}{}{{x}}^{{3}}{+}{e}{}{x}{+}{d}$ (2)
 > $A≔\mathrm{BezoutMatrix}\left(p,q,x\right)$
 ${A}{≔}\left[\begin{array}{ccc}{a}{}{f}{-}{c}{}{e}& {-}{b}{}{e}{-}{c}{}{d}& {-}{d}{}{b}\\ {b}{}{f}& {a}{}{f}{-}{c}{}{e}& {-}{d}{}{c}\\ {c}& {b}& {a}\end{array}\right]$ (3)
 > $\mathrm{Determinant}\left(A\right)$
 ${{a}}^{{3}}{}{{f}}^{{2}}{-}{2}{}{{a}}^{{2}}{}{c}{}{e}{}{f}{+}{a}{}{{b}}^{{2}}{}{e}{}{f}{+}{3}{}{a}{}{b}{}{c}{}{d}{}{f}{+}{a}{}{{c}}^{{2}}{}{{e}}^{{2}}{-}{{b}}^{{3}}{}{d}{}{f}{-}{b}{}{{c}}^{{2}}{}{d}{}{e}{+}{{c}}^{{3}}{}{{d}}^{{2}}$ (4)
 > $\mathrm{resultant}\left(p,q,x\right)$
 ${{a}}^{{3}}{}{{f}}^{{2}}{-}{2}{}{{a}}^{{2}}{}{c}{}{e}{}{f}{+}{a}{}{{b}}^{{2}}{}{e}{}{f}{+}{3}{}{a}{}{b}{}{c}{}{d}{}{f}{+}{a}{}{{c}}^{{2}}{}{{e}}^{{2}}{-}{{b}}^{{3}}{}{d}{}{f}{-}{b}{}{{c}}^{{2}}{}{d}{}{e}{+}{{c}}^{{3}}{}{{d}}^{{2}}$ (5)
 > $B≔\mathrm{BezoutMatrix}\left(p,q,x,\mathrm{method}=\mathrm{symmetric}\right)$
 ${B}{≔}\left[\begin{array}{ccc}{-}{f}{}{c}& {-}{b}{}{f}& {-}{a}{}{f}\\ {-}{b}{}{f}& {-}{a}{}{f}{+}{c}{}{e}& {d}{}{c}\\ {-}{a}{}{f}& {d}{}{c}& {-}{a}{}{e}{+}{b}{}{d}\end{array}\right]$ (6)
 > $\mathrm{MatrixOptions}\left(B,'\mathrm{shape}'\right)$
 $\left[{\mathrm{symmetric}}\right]$ (7)
 > $B≔\mathrm{BezoutMatrix}\left(p,q,x,\mathrm{method}=\mathrm{symmetric},\mathrm{shape}=\mathrm{rectangular}\right)$
 ${B}{≔}\left[\begin{array}{ccc}{-}{f}{}{c}& {-}{b}{}{f}& {-}{a}{}{f}\\ {-}{b}{}{f}& {-}{a}{}{f}{+}{c}{}{e}& {d}{}{c}\\ {-}{a}{}{f}& {d}{}{c}& {-}{a}{}{e}{+}{b}{}{d}\end{array}\right]$ (8)
 > $\mathrm{MatrixOptions}\left(B,'\mathrm{shape}'\right)$
 $\left[{}\right]$ (9)
 > $\mathrm{normal}\left(\frac{\mathrm{Determinant}\left(B\right)}{\mathrm{resultant}\left(p,q,x\right)}\right)$
 ${f}$ (10)