%"Klein
%Andre","A.A.B.Klein@uva.nl","\documentclass[12pt,fleqn]{article} \usepackage{amssymb} \usepackage[pc850]{inputenc} \usepackage{german} \usepackage{amsmath} \usepackage{amsfonts} \setcounter{MaxMatrixCols}{10} \pagestyle{empty} \voffset-1in \hoffset-1.1in \def\baselinestretch{0.9} \setlength{\oddsidemargin}{-0.5in} \setlength{\evensidemargin}{-0.5in} \setlength{\textwidth}{7.1in} \parindent0ex \topsep=4pt plus 1pt minus 3pt \oddsidemargin3.25cm \textwidth15cm \topmargin3.75cm \headheight0cm \headsep0cm \topskip0cm \textheight21cm \clubpenalty = 10000 \widowpenalty = 10000 \displaywidowpenalty = 10000 \sloppy \flushbottom \input{tcilatex} \begin{document}   \begin{center} {\large \textbf{Tensor Sylvester matrices and information matrices of multiple stationary processes}}  by  \textit{Andr\'{e} Klein}, Department of Quantitaive Economics, University of Amsterdam \\[0pt] Roetersstraat 11, 1018 WB Amsterdam, The Netherlands \\[0pt] \end{center}  \textbf{Abstract}  Consider the matrix polynomials $A(z)$ and $B(z)$ given by  $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A(z)=\dsum\limits_{j=0}^{p}A_{j}z^{j}$ and$\ B(z)=\dsum\limits_{j=0}^{q}B_{j}z^{j}$,  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.   \begin{thebibliography}{9} \bibitem{gohblerer} {\small {\large I.} \ {\large G}OHBERG, {\large L. L}% ERER, }Resultants of matrix polynomials. Bull. Amer. Math. Soc\textit{. }\  \textbf{82} {\small \ }(1976) 565-567.  \bibitem{kms} {\small {\large A. K}LEIN, {\large G. M}\textsc{\'{E}}LARD,  {\large P. S}PREIJ,} On the Resultant Property of the Fisher Information Matrix \ of a Vector ARMA process, Linear Algebra Appl. 403 (2005) 291-313. \end{thebibliography}  \end{document}","Multiple resultant matrix, Matrix Polynomial, Tensor Sylvester matrix, Fisher information matrix, VARMAX process","15A23","15A57","","08:05:54","Mon Mar 03 2008","145.18.180.139"
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