Bézout matrix

Examples

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

${\displaystyle B_3(f,g)=\left[\begin{array}{ccc}{u}_1{v}_0-u_0{v}_1& u_2{v}_0-u_0{v}_2& u_3{v}_0-u_0{v}_3\\ u_2{v}_0-u_0{v}_2& u_2{v}_1-u_1{v}_2+u_3{v}_0-u_0{v}_3& u_3{v}_1-u_1{v}_3\\ u_3{v}_0-u_0{v}_3& u_3{v}_1-u_1{v}_3& u_3{v}_2-u_2{v}_3\end{array}\right].}$

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

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

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 ${\displaystyle i=0,\dots ,n}$, either ${\displaystyle u_i}$ or ${\displaystyle v_i}$ is zero.

Properties

• ${\displaystyle B_n(f,g)}$ is symmetric (as a matrix);
• ${\displaystyle B_n(f,g)=-B_n(g,f)}$;
• ${\displaystyle B_n(f,f)=0}$;
• ${\displaystyle (f,g)\mapsto B_n(f,g)}$ is a bilinear function;
• ${\displaystyle B_n(f,g)}$ is a real matrix if f and g have real coefficients;
• ${\displaystyle B_n(f,g)}$ is nonsingular with ${\displaystyle n=max(\mathrm{deg}(f),\mathrm{deg}(g))}$ if and only if f and g have no common roots.
• ${\displaystyle B_n(f,g)}$ with ${\displaystyle n=max(\mathrm{deg}(f),\mathrm{deg}(g))}$ has determinant which is the resultant of f and g.
• Bezoutians can be reduced to block diagonal form with the help of confluent Vandermonde matrix.

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 ${\displaystyle B_n(p,q)}$. 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 ${\displaystyle B_n(p,q)}$ 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.