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, (Note that any coefficient u_i or v_i could be zero.) The Bézout matrix of order n associated with the polynomials f and g is where the entries b_{ij} result from the identity It is an n × n complex matrix, and its entries are such that if we let and for each, then: 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 (which is 4). The other zero entries are because for each, either u_i or v_i is zero.

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 denote r for the rank and σ for the signature of B_n(p,q). Then, we have the following statements: 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.

Citations