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. 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.
Let
f(z)
g(z)
f(z)=
n | |
\sum | |
i=0 |
uizi, g(z)=
n | |
\sum | |
i=0 |
vizi.
ui
vi
Bn(f,g)=\left(bij\right)i,j=0,...,n-1
bij
f(x)g(y)-f(y)g(x) | |
x-y |
n-1 | |
=\sum | |
i,j=0 |
bijxiyj.
It is an n × n complex matrix, and its entries are such that if we let
mij=min\{i,n-1-j\}
i,j=0,...,n-1
bij
mij | |
=\sum | |
k=0 |
(uj+k+1vi-k-ui-kvj+k+1).
To each Bézout matrix, one can associate the following bilinear form, called the Bézoutian:
\operatorname{Bez}:Cn x Cn\toC:(x,y)\mapsto\operatorname{Bez}(x,y)=x*Bn(f,g)y.
B3(f,g)=\left[\begin{matrix}u1v0-u0v1&u2v0-u0v2&u3v0-u0v3\\u2v0-u0v2&u2v1-u1v2+u3v0-u0v3&u3v1-u1v3\\u3v0-u0v3&u3v1-u1v3&u3v2-u2v3\end{matrix}\right].
f(x)=3x3-x
g(x)=5x2+1
B4(f,g)=\left[\begin{matrix}-1&0&3&0\\0&8&0&0\\3&0&15&0\\0&0&0&0\end{matrix}\right].
i=0,...,n
ui
vi
Bn(f,g)
Bn(f,g)=-Bn(g,f)
Bn(f,f)=0
(f,g)\mapstoBn(f,g)
Bn(f,g)
Bn(f,g)
n=max(\deg(f),\deg(g))
Bn(f,g)
n=max(\deg(f),\deg(g))
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
Bn(p,q)
Bn(p,q)
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.