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In [[mathematics]], a '''Bézout matrix''' (or '''Bézoutian''' or '''Bezoutiant''') is a special [[matrix (mathematics)#Square matrices and related definitions|square matrix]] associated with two [[polynomial]]s, introduced by {{harvs|txt|last=Sylvester|authorlink=James Joseph Sylvester|year=1853}} and {{harvs|txt|last=Cayley|year=1857|authorlink=Arthur Cayley}} and named after [[Étienne Bézout]]. Such matrices are sometimes used to test the [[stable polynomial|stability]] of a given polynomial.

==Definition==
Let ''f''(''z'') and ''g''(''z'') be two complex polynomials of degree at most ''n'' with coefficients (note that any coefficient could be zero):
:<math>f(z)=\sum_{i=0}^n u_i z^i,\quad\quad g(z)=\sum_{i=0}^n v_i z^i.</math>

The '''Bézout matrix''' of order ''n'' associated with the polynomials ''f'' and ''g'' is
:<math>B_n(f,g)=\left(b_{ij}\right)_{i,j=1,\dots,n}</math>
where the coefficients result from the identity
:<math>
\frac{f(x)g(y)-f(y)g(x)}{x-y}
=\sum_{i,j=1}^n b_{ij}\,x^{i-1}\,y^{j-1}.
</math>

It is in <math>\C^{n\times n}</math> and the entries of that matrix are such that if we note for each ''i'',''j''=1,...,n, <math>m_{ij}=\min\{i,n+1-j\}</math>, then:
:<math>b_{ij}=\sum_{k=1}^{m_{ij}}u_{j+k-1}v_{i-k}-u_{i-k}v_{j+k-1}.</math>

To each Bézout matrix, one can associate the following [[bilinear form]], called the Bézoutian:
:<math>\operatorname{Bez}:\C^n\times\C^n\to \C:(x,y)\mapsto \operatorname{Bez}(x,y)=x^*B_n(f,g)y.</math>

==Examples==
* For ''n''=3, we have for any polynomials ''f'' and ''g'' of degree (at most) 3:
:<math>B_3(f,g)=\left[\begin{matrix}u_1v_0-u_0 v_1 & u_2 v_0-u_0 v_2 & u_3 v_0-u_0 v_3\\u_2 v_0-u_0 v_2 & u_2v_1-u_1v_2+u_3v_0-u_0v_3 & u_3 v_1-u_1v_3\\u_3v_0-u_0v_3 & u_3v_1-u_1v_3 & u_3v_2-u_2v_3\end{matrix}\right].</math>

* Let <math>f(x)=3x^3-x</math> and <math>g(x)=5x^2+1</math> be two polynomials. Then:
:<math>B_4(f,g)=\left[\begin{matrix}-1 & 0 & 3 & 0\\0 &8 &0 &0 \\3&0&15&0\\0&0&0&0\end{matrix}\right].</math>
The last row and column are all zero as ''f'' and ''g'' have degree strictly less than ''n'' (equal 4). The other zero entries are because for each ''i''=0,...,n, either <math>u_i</math> or <math>v_i</math> is zero.

==Properties==
* <math>B_n(f,g)</math> is symmetric (as a matrix);
* <math>B_n(f,g)=-B_n(g,f)</math>;
* <math>B_n(f,f)=0</math>;
* <math>B_n(f,g)</math> is [[bilinear]]{{dn|date=December 2013}} in (''f'',''g'');
* <math>B_n(f,g)</math> is in <math>\mathbb{R}^{n\times n}</math> if ''f'' and ''g'' have real coefficients;
* <math>B_n(f,g)</math> is nonsingular with <math>n=max(deg(f),deg(g))</math> if and only if ''f'' and ''g'' have no common roots.
* <math>B_n(f,g)</math> with <math>n=max(deg(f),deg(g))</math> has [[determinant]] which is the [[resultant]] of ''f'' and ''g''.

==Applications==
An important application of Bézout matrices can be found in [[control theory]]. To see this, let ''f''(''z'') be a complex polynomial of degree ''n'' and denote by ''q'' and ''p'' the real polynomials such that ''f''(i''y'')=''q''(''y'')+i''p''(''y'') (where ''y'' is real). We also note ''r'' for the rank and ''σ'' for the signature of <math>B_n(p,q)</math>. Then, we have the following statements:
* ''f''(''z'') has ''n''-''r'' roots in common with its conjugate;
* the left ''r'' roots of ''f''(''z'') are located in such a way that:
** (''r''+''σ'')/2 of them lie in the open left half-plane, and
** (''r''-''σ'')/2 lie in the open right half-plane;
* ''f'' is [[stable polynomial|Hurwitz stable]] [[if and only if]] <math>B_n(p,q)</math> is [[Positive-definite matrix|positive definite]].

The third statement gives a necessary and sufficient condition concerning stability. Besides, the first statement exhibits some similarities with a result concerning [[Sylvester matrix|Sylvester matrices]] while the second one can be related to [[Routh-Hurwitz theorem]].

==References==

*{{Citation | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=Note sur la methode d’elimination de Bezout | year=1857 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002149818 | journal= J. Reine Angew. Math. | volume=53 | pages=366–367}}
*{{Citation | last2=Naĭmark | first2=M. A. | last1=Kreĭn | first1=M. G. | title=The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations | origyear=1936 | doi=10.1080/03081088108817420 | mr=638124 | year=1981 | journal=Linear and Multilinear Algebra | issn=0308-1087 | volume=10 | issue=4 | pages=265–308}}
* {{cite book |last1=Pan |first1=Victor |last2=Bini |first2=Dario |title=Polynomial and matrix computations |publisher=Birkhäuser |location=Basel, Switzerland |year=1994 |pages= |isbn=0-8176-3786-9 |oclc= |doi= }}
* {{cite book |last1=Pritchard |first1=Anthony J.|first2=Diederich |last2=Hinrichsen |title=Mathematical systems theory I: modelling, state space analysis, stability and robustness |publisher=Springer |location=Berlin |year=2005 |pages= |isbn=3-540-44125-5 |oclc= |doi= }}
*{{Citation | last1=Sylvester | first1=James Joseph | title=On a Theory of the Syzygetic Relations of Two Rational Integral Functions, Comprising an Application to the Theory of Sturm's Functions, and That of the Greatest Algebraical Common Measure | jstor=108572 | publisher=The Royal Society | year=1853 | journal=[[Philosophical Transactions of the Royal Society of London]] | issn=0080-4614 | volume=143 | pages= 407–548}}

{{DEFAULTSORT:Bezout matrix}}
[[Category:Polynomials]]
[[Category:Matrices]]

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