Bézout matrix

In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by James Joseph Sylvester in 1853 and Arthur Cayley in 1857 and named after Étienne Bézout.{{sfn|Sylvester|1853}}{{sfn|Cayley|1857}}

Bézoutian may also refer to the determinant of this matrix, which is equal to the resultant of the two polynomials. Bézout matrices are sometimes used to test the stability of a given polynomial.

Definition

Let $f(z)$ and $g(z)$ be two complex polynomials of degree at most n,

: $f(z) = \sum_{i=0}^n u_i z^i,\qquad g(z) = \sum_{i=0}^n v_i z^i.$

(Note that any coefficient $u_i$ or $v_i$ could be zero.) The Bézout matrix of order n associated with the polynomials f and g is

: $B_n(f,g)=\left(b_{ij}\right)_{i,j=0,\dots,n-1}$

where the entries $b_{ij}$ result from the identity

: $\frac{f(x)g(y)-f(y)g(x)}{x-y}
=\sum_{i,j=0}^{n-1} b_{ij}\,x^{i}\,y^{j}.$

It is an n × n complex matrix, and its entries are such that if we let $\ell_{ij} = \max\{0,i-j\}$ and $m_{ij} = \min\{i,n-1-j\}$ for each $i, j = 0, \dots, n-1$ , then:

: $b_{ij}=\sum_{k=\ell_{ij}}^{m_{ij}}(u_{j+k+1}v_{i-k}-u_{i-k}v_{j+k+1}).$

To each Bézout matrix, one can associate the following bilinear form, called the Bézoutian:

: $\operatorname{Bez}: \mathbb{C}^n\times\mathbb{C}^n \to \mathbb{C}: (x,y)\mapsto \operatorname{Bez}(x,y) = x^{*}B_n(f,g)\,y.$

Examples

For n = 3, we have for any polynomials f and g of degree (at most) 3:

:: $B_3(f,g)=\left[\begin{matrix}u_1v_0-u_0 v_1 & u_2 v_0-u_0 v_2 & u_3 v_0-u_0 v_3\\u_2 v_0-u_0 v_2 & u_2v_1-u_1v_2+u_3v_0-u_0v_3 & u_3 v_1-u_1v_3\\u_3v_0-u_0v_3 & u_3v_1-u_1v_3 & u_3v_2-u_2v_3\end{matrix}\right]\!.$

Let $f(x) = 3x^3-x$ and $g(x) = 5x^2+1$ be the two polynomials. Then:

:: $B_4(f,g)=\left[\begin{matrix}-1 & 0 & 3 & 0\\0 &8 &0 &0 \\3 & 0 & 15 & 0\\0 & 0 & 0 & 0\end{matrix}\right]\!.$

The last row and column are all zero as f and g have degree strictly less than n (which is 4). The other zero entries are because for each $i = 0, \dots, n$ , either $u_i$ or $v_i$ is zero.

Properties

$B_n(f,g)$ is symmetric (as a matrix);
$B_n(f,g) = -B_n(g,f)$ ;
$B_n(f,f) = 0$ ;
$(f, g) \mapsto B_n(f,g)$ is a bilinear function;
$B_n(f,g)$ is a real matrix if f and g have real coefficients;
$B_n(f,g)$ is nonsingular with $n=\max(\deg(f),\deg(g))$ if and only if f and g have no common roots.
$B_n(f,g)$ with $n = \max(\deg(f),\deg(g))$ 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(iy) = q(y) + ip(y) (where y is real). We also denote r for the rank and σ for the signature of $B_n(p,q)$ . 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 Hurwitz stable if and only if $B_n(p,q)$ is 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 matrices while the second one can be related to Routh–Hurwitz theorem.

Citations

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| doi=10.1515/crll.1857.53.366 }}
{{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 | orig-year=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 |isbn=0-8176-3786-9 }}
{{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 |isbn=3-540-44125-5 }}
{{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 | doi=10.1098/rstl.1853.0018| url=https://zenodo.org/record/1432412 | doi-access=free }}

Category:Polynomials

Category:Matrices (mathematics)