\documentclass[10pt, reqno]{amsart} \usepackage{lipsum} \usepackage[paperwidth=8.5in,paperheight=11in,left=1in,right=1in,top=0in,bottom=1in]{geometry} \usepackage{epsfig} \usepackage{color} \textheight 257mm \textwidth 173mm \topmargin -12mm \oddsidemargin -0mm \evensidemargin -0mm \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} \usepackage{blkarray} \usepackage{physics} \usepackage{amsmath} \usepackage{mathdots} \hypersetup{colorlinks=true,linkcolor=black, anchorcolor=green, citecolor=black, urlcolor=black, filecolor=magenta, pdftoolbar=true} \usepackage{fancyhdr} \textheight 22.5truecm \textwidth 14.5truecm \setlength{\oddsidemargin}{0.35in}\setlength{\evensidemargin}{0.35in} \usepackage{fontspec} \setmainfont{Times New Roman} \usepackage[utf8]{inputenc} \setlength{\topmargin}{-.5cm} \renewcommand{\phi}{\varphi} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{question}{Question} \newtheorem{proposition}{Proposition} \newtheorem{corollary}{Corollary} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem{example}{Example} \newtheorem{conclusion}{Conclusion} \newtheorem{conjecture}{Conjecture} \newtheorem{criterion}{Criterion} \newtheorem{summary}{Summary} \newtheorem{axiom}{Axiom} \newtheorem{problem}{Problem} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{exercise}{Exercise} \fancyhf{} \renewcommand{\headrulewidth}{1pt} \renewcommand{\headrule}{% \hrule width\headwidth height\headrulewidth\vskip2pt \hrule width\headwidth height\headrulewidth } \fancyhead[L]{\tiny MSC: Primary 15B05; Secondary 15A30.\\ } % \vspace*{1.5cm} \begin{document} \setcounter{page}{1} \title[ \large Product of Toeplitz Matrix And kth-order Slant Toeplitz Matrix.]{ \Large\bf Product of Toeplitz Matrix And kth-order Slant Toeplitz Matrix.} \author[\large S. Bensliman, M. Selmani, and B. Birech ]{ \bf\small S. Bensliman$^{1,*}$, M. Selmani$^{2,**}$, B. Birech$^{3,***}$.} \maketitle \thispagestyle{fancy} %\begin{center} %\large \bf Product of rectangular Toeplitz (Hankel) Matrices. %\end{center} %\vspace{18pt} %\begin{center} %\bf S. Bensliman$^{\it a*}$, A. Yagoub $^{\it a**}$, K. Toumache$^{\it b***}$ \vspace{12pt} %\end{center} \normalsize \rm \begin{center}\it $^{\it 1}$ Laboratoire de math\'ematiques pures et appliqu\'es. Universit\'e de Amar Telidji. Laghouat, 03000. Algeria. $^{\it 2}$ Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO) of University Oran 1, 31000. Algeria. $^{\it 3}$ Université IBN Khaldoun Tiaret, 31000. Algeria. $^{\it *}$ e–mail: s.bensliman@lagh-univ.dz\vspace{6pt} $^{\it **}$ e–mail: Mehdi.selmani@usto-univ.dz \vspace{6pt} $^{\it ***}$ e–mail: ibrahim.inzo.1996@gmail.com \vspace{6pt} \end{center} {\bf Abstract-}In this paper, we will study the product of a Toeplitz matrix with a kth-order slant Toeplitz matrix. We will find the necessary and sufficient conditions for this product to be a kth-order slant Toeplitz matrix. {\bf Keywords:} Toeplitz matrix, kth-order slant Toeplitz matrix, Hankel matrix, kth-order slant Hankel matrix. %\maketitle \vspace{12pt} \centerline{1. INTRODUCTION} %\label{sec1} \vspace{12pt} Some authors have deduced several properties of kth-order slant Toeplitz (or Hankel) matrices based on the kth-order slant Toeplitz operators, such as \cite{Bar}. In \cite{GU}, Gu gives necessary and sufficient conditions for a product of two Toeplitz matrices to be a Toeplitz matrix, and in \cite{BS}, Bensliman and Yagoub give necessary and sufficient conditions for a product of two rectangular Toeplitz matrices to be a rectangular Toeplitz matrix. The main objective of this paper is to study the product of two matrices $A$ and $B$, where $A$ is a Toeplitz matrix and $B$ is a kth-order slant Toeplitz matrix, and we find the conditions under which the product $AB$ is a kth-order slant Toeplitz matrix. It is easy to see that if $k=1$, we will find the necessary and sufficient conditions for a product of two Toeplitz matrices to be a Toeplitz matrix. Gu discussed this case in \cite{Gu}. \vspace{12pt} \centerline{2. PRELIMINARIES} %\label{sec1} \vspace{12pt} Let $M_{n\times n}(\mathbb{C)}$ denote the set of $n \times n$ matrices with entries in $\mathbb{C}$. Throughout this paper, the rows and columns of any $n\times n$ matrix will be indexed from $0$ to $n-1$. Accordingly, we denote the standard basis of $\mathbb{C}^{n}$ by $\lbrace e_{0},e_{1},...,e_{n-1}\rbrace$. Let $J$ and $S$ be two $n\times n $ matrices defined as follows \begin{center} $J=\left[ \begin{array}{cccc} 0 & 0 & \cdots &1 \\ & & 1 & 0 \\ & \iddots & & \vdots \\ & & & \\ 1 & & & 0 \end{array} \right],\qquad S=\left[ \begin{array}{ccccc} 0 & 0 & \cdots & \\ 1 & 0 & & \\ 0 & 1 & & \\ \vdots & & \ddots & \\ 0 & \cdots & & 1 & 0 \end{array} \right].$ \end{center} It is easy to verify that $J_{n}^{2}=I_{n}$ \newpage Recall that a Toeplitz matrix is a matrix of the form \begin{center} $A=\left[\begin{array}{ccccccc} a_{0}& \overline{\alpha}_{1} & \cdots & & &\overline{\alpha}_{n-1} \\ a_{1} & a_{0} & & & & \vdots \\ \vdots &\ddots & & & & \\ & & & & & \\ a_{n-1}& & & & & \end{array}\right],$ \end{center} and the kth-order slant Toeplitz matrix is a matrix of the form \begin{center} $B=\left[\begin{array}{ccccccc} b_{0}& b_{-1}& \cdots & & & b_{-n+1} \\ b_{k} & b_{k-1} & & & & b_{k-n+1} \\ b_{2k} & b_{2k-1} & & & & b_{2k-n+1} \\ \vdots & & & & & \\ & & & & & \\ b_{(n-1)k}& & & & & b_{(n-1)k-n+1} \end{array}\right],$ \end{center} where k is a positive integer such that $1\leq k\leq n-1$. In this work, the set of $n\times n$ Toeplitz matrices is denoted by $\mathcal{T}(n)$, and the set of $n\times n$ kth-order slant Toeplitz matrices is denoted by $\mathcal{T}(n,k)$. For two vectors $x$ and $y$ in $\mathbb{C}^{n}$ .the notation $x\otimes y$ denotes a tensor product of $x$ and $y$, which is defined by \begin{center} $(x\otimes y)z=\langle z,y\rangle x\qquad$ for all $z\in \mathbb{C}^{n},$ \end{center} where $x\otimes y=x\overline{y}^{T}$, and $\langle z,y\rangle =z_{1}\overline{y}_{1}+z_{1}\overline{y}_{2}+...+z_{n}\overline{y}_{n}.$ Let $\Delta A=A-SAS^{\ast k}$ be the displacement of matrix $A\in M_{n\times n}(\mathbb{C)}$. It is worth noting the following elementary lemma, which characterizes when a matrix in $ M_{n\times n}(\mathbb{C)}$ is a Toeplitz matrix or a kth-order slant Toeplitz matrix. \begin{lemma}\label{Lemma 2.1} Let $A$ be $n\times n$ matrix. Then \begin{enumerate} \item $A\in\mathcal{T}(n,k)$ if and only if $$\Delta A=\sum\limits_{i=0}^{k-1} u_{i}\otimes e_{i}+e_{0}\otimes v,$$ for some $u$ and $v$ be vectors in $\mathbb{C}^{n}$. \item $A\in\mathcal{T}(n)$ if and only if $$ A-SAS^{\ast}=u\otimes e_{0}+e_{0}\otimes v,$$ for some $u$ and $v$ be vectors in $\mathbb{C}^{n}$. \end{enumerate} \end{lemma} \begin{proof}[Proof] \begin{enumerate} \item Let $A$ be $n\times n$ Toeplitz matrix, then the matrix $S_{n}AS_{n}^{\ast k}$ is the same as matrix $A$ except for the k-1 first rows and the first column of the matrix $SAS^{\ast k}$, which are zero. Thus, $A-SAS^{\ast k}$ is $n\times n$ matrix all of its coefficients are zero except those in the k-1 first rows and the first column. \item The proof relies on the previous assertion for $k=1$. \end{enumerate} \end{proof} Throughout this paper, we assume that $a=(0,a_{1},a_{2},...,a_{n-1})^{T}$, $\alpha=(0,\alpha_{1},\alpha_{2},...,\alpha_{n-1})^{T}$, $\beta=(0,..,0,b_{-k},...,b_{-n+1})^{T}$, and $b_{(i,k)}=(b_{-i},b_{k-i},b_{2k-i},...,b_{(n-1)k-i})^{T}$, for $i=0,1,...,n-1$. \begin{lemma}\label{Lemma 2.3} Let $A\in\mathcal{T}(n)$ and $B\in\mathcal{T}(n,k)$. Then we have \begin{enumerate} \item[1)] $Ae_{0}=a+a_{0}e_{0}.$ \item[2)] $SAe_{n-1}= \hat{\alpha},$ where $\hat{\alpha}=(0,\overline{\alpha}_{n-1},...,\overline{\alpha}_{1})^{T}$. \item[3)] $S^{k}B^{\ast}e_{n-1}=b^{\sharp}$, where $b^{\sharp}=(0,...,\overline{b}_{(n-1)k},\overline{b}_{(n-1)k-1},...,\overline{b}_{(n-1)k-n+k+1})^{T}$. \end{enumerate} \end{lemma} \begin{lemma}\label{L1} Let $x_{1},x_{2},y_{1}$ and $y_{2}$ be vectors in $\mathbb{C}^{n}$ . If the matrices $x_{1}\otimes y_{1}$ and $x_{2}\otimes y_{2}$ are equal, then one of the following cases is satisfied \begin{enumerate} \item[1)] $x_{1}=x_{2}=0$ or $y_{1}=y_{2}=0$. \item[2)] $x_{1}=y_{2}=0$ or $y_{1}=x_{2}=0$. \item[3)] $x_{1}= \lambda x_{2} $ and $\overline{\lambda}y_{1}=y_{2}$ for some nonzero scalar $\lambda$. \end{enumerate} By convention $\frac{1}{\infty}=0$, we can propose that $x_{1}\otimes y_{1}=x_{2}\otimes y_{2}$, then one of the following cases is satisfied \begin{enumerate} \item[i.] Trivial case: $x_{1}=x_{2}=0$ or $y_{1}=y_{2}=0$. \item[ii.] $x_{1}= \lambda x_{2} $ and $\overline{\lambda}y_{1}=y_{2}$ for some scalar $\lambda\in\hat{\mathbb{C}}=\mathbb{C}\cup\lbrace \infty\rbrace$. \end{enumerate} \end{lemma} The following lemma plays a pivotal role in our analysis, as it facilitates the manipulation of the product between a Toeplitz matrix and a kth-order slant Toeplitz matrix. \begin{lemma}\label{Lemma 2.4} Let $A\in\mathcal{T}(n)$ and $B\in\mathcal{T}(n,k)$. Then we have $$\Delta (AB)=\sum\limits_{i=0}^{k-1} Ab_{(i,k)}\otimes e_{i}+a\otimes \beta +e_{0}\otimes (\overline{a_{0}}\beta +S^{k}B^{\ast}S^{\ast }\alpha) -\hat{\alpha}\otimes b^{\sharp}.$$ \end{lemma} \vspace{12pt} \centerline{3.PRODUCT OF TOEPLITZ MATRIX AND kTH-ORDER SLANT TOEPLITZ MATRIX.} %\label{sec1} \vspace{12pt} \begin{theorem}\label{coro 3.3} Let $A\in\mathcal{T}(n)(A\neq 0)$ and $B\in\mathcal{T}(n,k)(B\neq 0)$ with $1\leq k \leq n-1$. Then $AB\in\mathcal{T}(n,k)$ if and only if one of the following conditions is met \begin{enumerate} \item[1)] $A=a_{0}I$, for some $a_{0}\in\mathbb{C}$. \item[2)]$A$ is an upper triangular Toeplitz matrix and $B$ is kth-order slant Toeplitz matrix with $ b^{\sharp}=0$. \item[3)] $A$ is an lower triangular Toeplitz matrix and $B$ is kth-order slant Toeplitz matrix with $ \beta=0$. \item[4)] $B$ is kth-order slant Toeplitz matrix with $\beta=b^{\sharp}=0$. \item[5)] $A$ is a Toeplitz matrix with $a=\lambda \hat{\alpha}$, and $B$ is kth-order slant Toeplitz matrix with $b^{\sharp}=0\overline{\lambda}\beta$, for some $\lambda \in\mathbb{C}^{\ast}.$ \end{enumerate} \end{theorem} \centerline{ACKNOWLEDGMENTS} %\label{sec1} \vspace{2pt} This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria. \renewcommand{\refname}{\centerline{\small REFERENCES}} \bibliographystyle{amsplain} \begin{thebibliography}{99} \bibitem{BS} Bensliman, S., Yagoub, A., Toumache, K. Product of rectangular Toeplitz (Hankel) Matrices, Comput. Math. Math. Phys, vol 64, 11, 2510–2522 (2024). \bibitem{GU} Gu, C., Patton, L. “Commutation Relations for Toeplitz and Hankel Matrices," SIAM J. matrix anal.appl. 24(3), 728-746 (2003). \bibitem{Bar} Łanucha, B., Michalska, M. Compressions of kth-order slant Toeplitz operators to model spaces, Lithuanian Mathematical Journal, vol 62, 1, 69–87 (2021). \bibitem{YA} Yagoub, A. “Products of asymmetric truncated Toeplitz operators." Adv. Oper. Theory 5, 233-247 (2020). \end{thebibliography} \clearpage %\caption{The part of the experimental spectrum of Be$_2$ from \cite{Bondybey}.}\label{exp} \end{document}