Bézout matrix

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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[edit]

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[edit]

  • 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[edit]

Applications[edit]

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[edit]