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%Klein
%Andre","A.A.B.Klein@uva.nl","
\section{Tensor Sylvester matrices and information matrices of multiple stationary processes} By {\sl Andr\'{e} Klein, Department of Quantitaive Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands}.  

\medskip  \noindent Consider the matrix polynomials $A(z)$ and $B(z)$ given by
$$A(z)=\sum_{j=0}^{p}A_{j}z^{j}$$ and $$\ B(z)=\sum_{j=0}^{q}B_{j}z^{j},$$ 
	 where $A_{0}\equiv B_{0}\equiv I_{n}$.\newline Gohberg and Lerer [1] study the resultant property of the tensor Sylvester matrix $\mathcal{S}^{\otimes }(-B,A)\triangleq \mathcal{S}(-B\otimes I_{n},I_{n}\otimes A)$ or $\mathcal{S}^{\otimes }(-B,A)=\left(  \begin{array}{ccccccc} \left( -I_{n}\right) \otimes I_{n} & \left( -B_{1}\right) \otimes I_{n} &  \cdots & \left( -B_{q}\right) \otimes I_{n} & 0_{n^{2}\times n^{2}} & \cdots & 0_{n^{2}\times n^{2}} \\  0_{n^{2}\times n^{2}} & \ddots & \ddots &  & \ddots & \ddots & \vdots \\  \vdots & \ddots & \ddots & \ddots &  & \ddots & 0_{n^{2}\times n^{2}} \\  0_{n^{2}\times n^{2}} & \cdots & 0_{n^{2}\times n^{2}} & \left( -I_{n}\right) \otimes I_{n} & \left( -B_{1}\right) \otimes I_{n} & \cdots &  \left( -B_{q}\right) \otimes I_{n} \\  I_{n}\otimes I_{n} & I_{n}\otimes A_{1} & \cdots & I_{n}\otimes A_{p} &  0_{n^{2}\times n^{2}} & \cdots & 0_{n^{2}\times n^{2}} \\  0_{n^{2}\times n^{2}} & \ddots & \ddots &  & \ddots & \ddots & \vdots \\  \vdots & \ddots & \ddots & \ddots &  & \ddots & 0_{n^{2}\times n^{2}} \\  0_{n^{2}\times n^{2}} & \cdots & 0_{n^{2}\times n^{2}} & I_{n}\otimes I_{n} & I_{n}\otimes A_{1} & \cdots & I_{n}\otimes A_{p} \end{array} \right) $.  In [1] it is proved that the matrix polynomials $A(z)$ and $B(z)$ have at least one common eigenvalue if and only if det$\mathcal{S}^{\otimes }(-B,A)=0 $ or when the matrix $\mathcal{S}^{\otimes }(-B,A)$ is singular$.$ In other words, the tensor Sylvester matrix $\mathcal{S}^{\otimes }(-B,A)$ becomes singular if and only if the scalar polynomials det $A(z)=0$ and det $B(z)=0$ have at least one common root. Consequently, it is a multiple resultant.  In [2], this property is extended to the Fisher information matrix of a stationary vector autoregressive and moving average process, VARMA process.  The purpose of this talk consists of displaying a representation of the Fisher information matrix of a stationary VARMAX process in terms of tensor Sylvester matrices, the X stands for exogenous or control variable. The VARMAX process is of common use in stochastic systems and control.   
%Multiple resultant matrix, Matrix Polynomial, Tensor Sylvester matrix, Fisher information matrix, VARMAX process","15A23","15A57","","08:19:16","Mon Mar 03 2008","145.18.180.139"
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