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.
[1][2] 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.
be two complex polynomials of degree at most n, (Note that any coefficient
The Bézout matrix of order n associated with the polynomials f and g is where the entries
result from the identity It is an n × n complex matrix, and its entries are such that if we let
, then: To each Bézout matrix, one can associate the following bilinear form, called the Bézoutian: The last row and column are all zero as f and g have degree strictly less than n (which is 4).
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
Besides, the first statement exhibits some similarities with a result concerning Sylvester matrices while the second one can be related to Routh–Hurwitz theorem.