In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Bézout matrix (or Bézoutian) is a special square matrix associated with two polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s. Such matrices are sometimes used to test the stability

Stable polynomial

A polynomial is said to be stable if either:* all its roots lie in the open left half-plane, or* all its roots lie in the open unit disk.The first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...

 of a given polynomial.

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 and the entries of that matrix are such that if we note for each i,j=1,...,n, , then:

To each Bézout matrix, one can associate the following bilinear form, called the Bézoutian:

Examples

• For n=3, we have for any polynomials f and g of degree (at most) 3:

• Let and be two polynomials. Then:

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 i=0,...,n, either or is zero.

Properties

• is symmetric (as a matrix);
• ;
• ;
• is bilinear
  Bilinear
  Bilinear may refer to:* Bilinear sampling, a method in computer graphics for choosing the color of a texture* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...
  
   in (f,g);
• is in if f and g have real coefficients;
• is nonsingular with if and only if f and g have no common roots.
• with has determinant
  Determinant
  In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
  
   which is the resultant of f and g.

Applications

An important application of Bézout matrices can be found in control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

. 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 . Then, we have the following statements:

• f(z) has n-r roots in common with its conjugate;
• the left r roots of f(z) are located in such a way that:
  • (r+σ)/2 of them lie in the open left half-plane, and
  • (r-σ)/2 lie in the open right half-plane;
• f is Hurwitz stable
  Stable polynomial
  A polynomial is said to be stable if either:* all its roots lie in the open left half-plane, or* all its roots lie in the open unit disk.The first condition defines Hurwitz stability and the second one Schur stability. Stable polynomials arise in various mathematical fields, for example in...
  
   if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
    is positive definite
  Positive-definite matrix
  In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
  
  .

The third statement gives a necessary and sufficient condition concerning stability. Besides, the first statement exhibits some similarities with a result concerning Sylvester matrices

Sylvester matrix

In mathematics, a Sylvester matrix is a matrix associated to two polynomials that provides information about those polynomials. It is named for James Joseph Sylvester.-Definition:...

 while the second one can be related to Routh-Hurwitz theorem.