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. 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.
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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):
The Bézout matrix of order n associated with the polynomials f and g is
where the coefficients result from the identity
It is in
To each Bézout matrix, one can associate the following bilinear form, called the Bézoutian:
Examples
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
Properties
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
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.