Bézout matrix

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In mathematics, a Bézout matrix (or Bézoutian) is a special square matrix associated to two polynomials. Such matrices are sometimes used to test the stability of a given polynomial.

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[edit] 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 to the polynomials f and g is

Bn(f,g) = [b]ij.

It is in \mathbb{C}^{n\times n} and the entries of that matrix are such that if we note for each i,j=1,...,n, mij = 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:

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

[edit] 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) = 3x3 − x and g(x) = 5x2 + 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 due to the fact that for each i=0,...,n, either ui or vi is zero.

[edit] Properties

  • Bn(f,g) is symmetric (as a matrix);
  • Bn(f,g) = − Bn(g,f);
  • Bn(f,f) = 0;
  • Bn(f,g) is bilinear in (f,g);
  • Bn(f,g) is in \mathbb{R}^{n\times n} if f and g have real coefficients;
  • Bn(f,g) is nonsingular with n = max(deg(f),deg(g)) if and only if f and g have no common roots.
  • Bn(f,g) with n = max(deg(f),deg(g)) has determinant which is the resultant of f and g.

[edit] 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 Bn(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 iff Bn(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.

[edit] References

  • D. Hinrichsen and A.J. Pritchard, Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness, Springer-Verlag, Berlin-Heidelberg, 2005