User:Cturnes/Anderson-Jury Bezoutian

An Anderson-Jury Bézoutian is a generalized form of the scalar Bézout matrix (or Bézoutian) that arises from the coefficients of polynomial matrices rather than univariate polynomials. The name Anderson-Jury is attributed to the authors of the seminal paper which first introduced the generalized Bezoutian form. They have been studied due to their connection with the stability of matrix polynomials and for their role in control theory. They are also of interest for their role in the inversion of block Hankel matrices.

Definition
Let the quadruple $$\{A(z),B(z),C(z),D(z)\}$$ be a set of four polynomial matrices. The Anderson-Jury Bézoutian form associated with the quadruple is given by:
 * $$ \Gamma(x,y)=\frac{1}{x-y}\left[A(x)D(y)-B(x)C(y)\right].$$

Using the original definition supplied by B.D.O. Anderson and E.I. Jury, the polynomial matrices have the additional constraint that:
 * $$ A(x)D(x)=B(x)C(x), $$

which is necessary and sufficient for the polynomial form $$\Gamma(x,y)$$ to be integral in $$x,y$$. With this constraint, a real rational function of $$z$$ can be defined as $$W(z)=A(z)^{-1}B(z)=D(z)C(z)^{-1}$$. If $$A(z)$$ is of degree $$n$$, $$C(z)$$ is of degree $$m$$, $$D(z)$$ is of degree $$l$$, where $$l\leq m$$, then the Anderson-Jury Bézoutian of $$\{A(z),B(z),C(z),D(z)\}$$ may also be expressed as:
 * $$ \Gamma(x,y)=\sum_{i=1}^n\sum_{j=1}^m \Gamma_{ij}x^{i-1}y^{j-1},$$

with
 * $$ \Gamma_{ij}=\sum_{k\geq 0} (A_{i+k}D_{j-1-k}-B_{i+k}C_{j-1-k})=\sum_{k\geq 0} (B_{i-k-1}C_{j+k}-A_{i-k-1}C_{j-k}). $$

Properties

 * Unlike the scalar Bézoutian matrix, the Anderson-Jury Bézoutian is not in general symmetric.
 * For any block Hankel matrix $$H$$, there exists an Anderson-Jury Bézoutian which is a reflexive generalized inverse of $$H$$. For nonsingular $$H$$, the inverse of $$H$$ is an Anderson-Jury Bézoutian, as the reflexive generalized inverse is identically equal to the inverse.

Generalizations
Wimmer introduced the following, more general form of the Anderson-Jury Bézoutian : For a fixed field $$\mathbb{K}$$ let $$W\in \mathbb{K}^{p\times q}(z)$$ be a strictly proper rational function. That is, let $$W$$ have the form
 * $$W(z) = \sum_{i>0} W_iz^{-i}, $$

where each $$W_i$$ is a $$p\times q$$ matrix with entries drawn from $$\mathbb{K}$$. Additionally, define the polynomial matrices
 * $$ A\in \mathbb{K}^{p\times p}[z],\ C\in \mathbb{K}^{q\times q}[z],\ B,D\in \mathbb{K}^{p\times q}[z] $$

such that
 * $$ A[z]=A_0+A_1z+\cdots+A_nz^n,\ \ C[z]=C_0+C_1z+\cdots+C_mz^m $$

are nonsingular and
 * $$\pi_-(A^{-1}B)=\pi_-(DC^{-1})=W.$$

Here, $$\pi_-$$ is a projection operator that selects the strictly proper portion of a rational function (see Fuhrmann 1996, Chapter 1, Section 3.4). Then the generalized Anderson-Jury Bézoutian $$B=B(A,B,C,D)$$ of the quadruple $$(A,B,C,D)$$ is the $$rp\times sq$$ matrix $$B=[B_{ij}],\ i=1,\ldots,r,\ j=1,\ldots, s$$, where the block entries $$B_{ij}$$ are given by the following equation:
 * $$\Delta(x,y)=A(x)\frac{W(y)-W(x)}{x-y}C(y)=\sum_{i=1}^r\sum_{j=1}^s B_{ij}x^{i-1}y^{j-1}.$$

In the standard Anderson-Jury Bézoutian, there is the additional assumption that
 * $$\pi_-(A^{-1}B)=A^{-1}B\ \mathrm{and}\ \pi_-(DC^{-1})=DC^{-1},$$

in which case $$\Delta$$ becomes
 * $$ \Delta(x,y)=\frac{A(x)D(y)-B(x)C(y)}{x-y}=\Gamma(x,y).$$