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\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}

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%\caption{The part of the experimental spectrum of Be$_2$ from \cite{Bondybey}.}\label{exp}	

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