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In mathematics, a Bézout matrix (or Bézoutian or Bezoutiant) is a special
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
associated with two
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s, introduced by and and named after
Étienne Bézout Étienne Bézout (; 31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France. Work In 1758 Bézout was elected an adjoint in mechanics of the F ...
. Bézoutian may also refer to the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
of this matrix, which is equal to the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over ...
of the two polynomials. Bézout matrices are sometimes used to test the
stability Stability may refer to: Mathematics * Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural st ...
of a given polynomial.


Definition

Let f(z) and g(z) be two
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
polynomials of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
at most ''n'', :f(z) = \sum_^n u_i z^i,\qquad g(z) = \sum_^n v_i z^i. (Note that any
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
u_i or v_i could be zero.) The Bézout matrix of order ''n'' associated with the polynomials ''f'' and ''g'' is :B_n(f,g)=\left(b_\right)_ where the entries b_ result from the identity : \frac =\sum_^ b_\,x^\,y^. It is an ''n'' × ''n'' complex matrix, and its entries are such that if we let m_ = \min\ for each i, j = 0, \dots, n-1, then: :b_=\sum_^(u_v_-u_v_). To each Bézout matrix, one can associate the following
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is lin ...
, called the Bézoutian: :\operatorname: \mathbb^n\times\mathbb^n \to \mathbb: (x,y)\mapsto \operatorname(x,y) = x^B_n(f,g)\,y.


Examples

* For ''n'' = 3, we have for any polynomials ''f'' and ''g'' of degree (at most) 3: ::B_3(f,g)=\left beginu_1v_0-u_0 v_1 & u_2 v_0-u_0 v_2 & u_3 v_0-u_0 v_3\\u_2 v_0-u_0 v_2 & u_2v_1-u_1v_2+u_3v_0-u_0v_3 & u_3 v_1-u_1v_3\\u_3v_0-u_0v_3 & u_3v_1-u_1v_3 & u_3v_2-u_2v_3\end\right!. * Let f(x) = 3x^3-x and g(x) = 5x^2+1 be the two polynomials. Then: ::B_4(f,g)=\left begin-1 & 0 & 3 & 0\\0 &8 &0 &0 \\3 & 0 & 15 & 0\\0 & 0 & 0 & 0\end\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, \dots, n, either u_i or v_i is zero.


Properties

* B_n(f,g) is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
(as a matrix); * B_n(f,g) = -B_n(g,f); * B_n(f,f) = 0; * (f, g) \mapsto B_n(f,g) is a bilinear function; * B_n(f,g) is a real matrix if ''f'' and ''g'' have
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
coefficients; * B_n(f,g) is nonsingular with n=\max(\deg(f),\deg(g)) if and only if ''f'' and ''g'' have no common
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
. * B_n(f,g) with n = \max(\deg(f),\deg(g)) has
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
which is the
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over ...
of ''f'' and ''g''.


Applications

An important application of Bézout matrices can be found in
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
. 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''(i''y'') = ''q''(''y'') + i''p''(''y'') (where ''y'' is real). We also denote ''r'' for the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
and ''σ'' for the signature of B_n(p,q). 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
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
B_n(p,q) is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * 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

* * * * * {{DEFAULTSORT:Bezout matrix Polynomials Matrices