Bézout matrix

In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special square matrix associated with two polynomials, introduced by Sylvester (1853) and Cayley (1857) and named after Étienne Bézout. Such 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 with coefficients (note that any coefficient could be zero):

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

The Bézout matrix of order n associated with the polynomials f and g is

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

where the coefficients result from the identity

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

It is in $\C^{n\times n}$ and the entries of that matrix are such that if we note for each i,j=1,...,n, $m_{ij}=\min\{i,n+1-j\}$ , then:

b_{ij}=\sum_{k=1}^{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}:\C^n\times\C^n\to \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 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 (equal 4). The other zero entries are because for each i=0,...,n, either $u_i$ or $v_i$ is zero.

Properties

$B_n(f,g)$ is symmetric (as a matrix);
$B_n(f,g)=-B_n(g,f)$ ;
$B_n(f,f)=0$ ;
$B_n(f,g)$ is bilinear in (f,g);
$B_n(f,g)$ is in $\mathbb{R}^{n\times n}$ 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 note 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.

References

Cayley, Arthur (1857), "Note sur la methode d’elimination de Bezout", J. Reine Angew. Math. 53: 366–367
Kreĭn, M. G.; Naĭmark, M. A. (1981) [1936], "The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations", Linear and Multilinear Algebra 10 (4): 265–308, doi:10.1080/03081088108817420, ISSN 0308-1087, MR 638124
Pan, Victor; Bini, Dario (1994). Polynomial and matrix computations. Basel, Switzerland: Birkhäuser. ISBN 0-8176-3786-9.
Pritchard, Anthony J.; Hinrichsen, Diederich (2005). Mathematical systems theory I: modelling, state space analysis, stability and robustness. Berlin: Springer. ISBN 3-540-44125-5.
Sylvester, James Joseph (1853), "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", Philosophical Transactions of the Royal Society of London (The Royal Society) 143: 407–548, doi:10.1098/rstl.1853.0018, ISSN 0080-4614, JSTOR 108572