Bézout matrix explained

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.

Definition

Let

f(z)

and

g(z)

be two complex polynomials of degree at most n,

f(z)=

n
\sum
i=0

uizi,    g(z)=

n
\sum
i=0

vizi.

(Note that any coefficient

ui

or

vi

could be zero.) The Bézout matrix of order n associated with the polynomials f and g is

Bn(f,g)=\left(bij\right)i,j=0,...,n-1

where the entries

bij

result from the identity
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\}

for each

i,j=0,...,n-1

, then:

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.

Examples

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

and

g(x)=5x2+1

be the two polynomials. Then:

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].

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

i=0,...,n

, either

ui

or

vi

is zero.

Properties

Bn(f,g)

is symmetric (as a matrix);

Bn(f,g)=-Bn(g,f)

;

Bn(f,f)=0

;

(f,g)\mapstoBn(f,g)

is a bilinear function;

Bn(f,g)

is a real matrix 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.

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

Bn(p,q)

. Then, we have the following statements:

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.

References