\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \usepackage{url,amsbsy,amsopn,amstext,amsxtra,euscript,amscd} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\Large\bf On The Pfaffians and Determinants of Some \\ \vskip .1in Skew-Centrosymmetric Matrices } \vskip 1cm Fatih Y\i lmaz\\ Polatl\i\ Art and Science Faculty \\ Gazi University\\ 06500 Teknikokullar / Ankara \\ Turkey\\ \href{mailto:fatihyilmaz@gazi.edu.tr}{\tt fatihyilmaz@gazi.edu.tr} \\ \ \\ Tomohiro Sogabe\\ Graduate School of Engineering\\ Nagoya University \\ Furo-cho, Chikusa-ku \\ Nagoya, 464-8601 \\ Japan\\ \href{mailto:sogabe@na.cse.nagoya-u.ac.jp}{\tt sogabe@na.cse.nagoya-u.ac.jp} \\ \ \\ Emrullah K\i rklar\\ Polatl\i\ Art and Science Faculty \\ Gazi University \\ 06500 Teknikokullar / Ankara \\ Turkey\\ \href{mailto:e.kirklar@gazi.edu.tr}{\tt e.kirklar@gazi.edu.tr} \\ \end{center} \vskip .2 in \def\cI{{\mathcal I}} \def\cR{{\mathcal R}} \def\a{{\alpha}} \def\b{{\beta}} \def\g{{\gamma}} \def\d{{\delta}} \def\l{{\lambda}} \def\o{{\omega}} \def\e{{\epsilon}} \def\ep{{\varepsilon}} \def\s{{\sigma}} \def\t{{\tau}} \def\v{{\nu}} \def\th{{\theta}} \def \K{{\bbbk}} \def\E{{\mathbf E}} \def\G{{\mathcal G}} \def\O{{\mathcal O}} \def \R{{\bbbr}} \def\({\left(} \def\){\right)} \def\lf{\lfloor} \def\rf{\rfloor} \def\lc{\lceil} \def\rc{\rceil} \begin{abstract} This paper shows that the Pfaffians and determinants of some skew centrosymmetric matrices can be computed by a paired two-term recurrence relation, or a general number sequence of second order. As a result, the complexities of the formulas are of order $n$. Furthermore, the formulas have no divisions at all, i.e., they fall into the class of breakdown-free algorithms. \end{abstract} \section{Introduction} The \textit{determinant\ }is one of the basic parameters in matrix theory. The \textit{determinant} of a square matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}$ is defined as \begin{equation*} \det (A)=\underset{\sigma \in S_{n}}{\sum }\mathrm{sgn}(\sigma )\underset{i=1}{\overset{n}{\prod }}a_{i,\sigma (i)}, \end{equation*} where the symbol $S_{n}$ denotes the group of permutations of sets with $n$ elements and the symbol sgn$(\sigma)$ denotes the signature of $\sigma \in S_n$. The \textit{Pfaffian} of a skew-symmetric matrix $A=(a_{i,j}) \in \mathbb{C}^{2k\times 2k}$ is defined by \begin{equation} \label{Def:Pfaffian} \mathrm{Pf}(A)=\frac{1}{2^k k!}\underset{\sigma \in S_{2k}}{\sum }\mathrm{sgn}(\sigma )\underset{i=1}{\overset{k}{\prod }}a_{\sigma (2i-1),\sigma (2i)}, \end{equation} and is closely related to the determinant. In fact, Cayley's theorem states that the square of the Pfaffian of a matrix is equal to the determinant of the matrix, i.e., \begin{equation*} \det (A)=\mathrm{Pf}(A)^{2}. \end{equation*} Matrix $A$ is called a {\it centrosymmetric} matrix if $A=JAJ^{-1}$, where $J$ is the anti-diagonal matrix whose anti-diagonal elements are one with all others being zero. If $A=-JAJ^{-1}$, the matrix is said to be {\it skew-centrosymmetric}. Skew-centrosymmetric matrices arise in many fields of science including numerical solutions of certain differential equations, digital signal processing, information theory, statistics, linear systems theory, and some Markov processes \cite{1,2,3,4,5,6}. In general, the complexities of the Pfaffian and the determinant are of the order $\mathcal{O}(n^3)$. This paper describes efficient computational formulas for the Pfaffians and determinants of special matrices for which the complexities of the formulas are of the order $\mathcal{O}(n)$. The formulas have no divisions at all, i.e., the formulas fall into the class of breakdown-free algorithms. \section{Pfaffians of skew-centrosymmetric matrices} \begin{definition} $A_{n}=(a_{i,j})$ and $B_{n}=(b_{i,j})$ denote $n$-by-$n$ matrices with the following elements: \begin{align*} &a_{i,j}= \begin{cases} a, & \text{if $j=i+1$;} \\ -a, & \text{if $i=j+1$;} \\ 0, & \text{otherwise,} \end{cases} \\[1em] &b_{i,j}= \begin{cases} (-1)^{i+1}b, & \text{if $i+j=n+1$;} \\ 0, & \text{otherwise,} \end{cases} \end{align*} where $1\leq i,j\leq n.$ \end{definition} \begin{definition}\label{def:F} $\mathcal{F}_{n}$ and $\mathcal{G}_{n}$ denote $2$-by-$2$ block matrices of the following form: \begin{equation*} \mathcal{F}_{n}= \left( \begin{array}{cc} A_{k} & B_{k} \\ (-1)^{k}B_{k} & A_{k} \end{array} \right), \quad \mathcal{G}_{n}= \left( \begin{array}{cc} A_{k} & -B_{k} \\ (-1)^{k+1}B_{k} & A_{k} \end{array} \right), \end{equation*} where $n=2k$. \end{definition} For example, if $n=10$, it follows from Definition \ref{def:F} that \begin{align*} \mathcal{F}_{10} &=\left( \begin{array}{cc} A_{5} & B_{5} \\ (-1)^{5}B_{5} & A_{5} \end{array} \right)\\[1em] &=\left( \begin{array}{ccccc|ccccc} 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b \\ -a & 0 & a & 0 & 0 & 0 & 0 & 0 & -b & 0 \\ 0 & -a & 0 & a & 0 & 0 & 0 & {b} & 0 & 0 \\ 0 & 0 & -a & 0 & a & 0 & -b & 0 & 0 & 0 \\ 0 & 0 & 0 & -a & 0 & b & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & -b & 0 & a & 0 & 0 & 0 \\ 0 & 0 & 0 & b & 0 & -a & 0 & a & 0 & 0 \\ 0 & 0 & {-b} & 0 & 0 & 0 & -a & 0 & a & 0 \\ 0 & b & 0 & 0 & 0 & 0 & 0 & -a & 0 & a \\ -b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -a & 0% \end{array} \right) . \end{align*} We now describe algorithms for computing the Pfaffians of $\mathcal{F}_{n}$ and $\mathcal{G}_{n}$. \begin{theorem} Let $\{f_n\}$ and $\{g_{n}\}$ be the recursively defined sequences below: \begin{align*} &f_{n}=bg_{n-1}+a^{2}f_{n-2}\text{ \ \ for \ }f_{1}=b, \\ &g_{n}=-bf_{n-1}+a^{2}g_{n-2}\text{ \ \ for \ }g_{1}=-b. \end{align*} Then, for $n=2k$, we obtain \begin{equation*} f_{k}=\mathrm{Pf}(\mathcal{F}_{n})\text{ \ \ and \ \ }g_{k}=\mathrm{Pf}(% \mathcal{G}_{n}), \end{equation*} where $f_{-1}=0,$ $f_{0}=1$ and $g_{-1}=0,$ $g_{0}=1.$ \end{theorem} \begin{proof} The proof is done by induction on $k$. For $k=1$, \begin{equation*} \mathcal{F}_{2}=\left( \begin{array}{cc} A_{1} & B_{1} \\ -B_{1} & A_{1} \end{array} \right) =\left( \begin{array}{cc} 0 & b \\ -b & 0 \end{array} \right) \text{ and } \mathcal{G}_{2}=\left( \begin{array}{cc} A_{1} & -B_{1} \\ B_{1} & A_{1} \end{array} \right) =\left( \begin{array}{cc} 0 & -b \\ b & 0 \end{array} \right). \end{equation*} The definition of the Pfaffian in \eqref{Def:Pfaffian} clearly indicates that $\mathrm{Pf}(\mathcal{F}_{2})=b$ and $\mathrm{Pf}(\mathcal{G}_{2})=-b$. Thus, $f_{1}=b=\mathrm{Pf}(\mathcal{F}_{2}), g_{1}=-b=\mathrm{Pf}(\mathcal{G}_{2}).$ Let us assume that the recurrence relations hold for all $t\leq k$. Then we show that they hold for $k=t+1$. \begin{align} \mathcal{F}_{2t+2} & =\left( \begin{array}{c|c} A_{t+1} & B_{t+1} \\ \hline (-1)^{t+1}B_{t+1} & A_{t+1} \end{array} \right) \nonumber \\[1em] &=\left( \begin{tabular}{c|cccc|c} $0$ & $a$ & $0$ & $\cdots $ & $0$ & $b$ \\ \hline $-a$ & & & & & $0$ \\ $0$ & & $A_{t}$ & $-B_{t}$ & & $\vdots $ \\ $\vdots $ & & $(-1)^{t+1}B_{t}$ & $A_{t}$ & & $0$ \\ $0$ & & & & & $a$ \\ \hline $-b$ & $0$ & $\cdots $ & $0$ & $-a$ & $0$ \end{tabular} \right). \label{10} \end{align} From the expansion formula along with $2t+2$ column of (\ref{10}), it follows that \begin{equation} \mathrm{Pf}(\mathcal{F}_{2t+2})=b\mathrm{Pf}(\mathcal{G}_{2t})+a\mathrm{Pf}(% \mathcal{M}_{2t})=bg_{t}+a\mathrm{Pf}(\mathcal{M}_{2t}), \label{20} \end{equation} where \begin{equation} \mathcal{M}_{2t}=\left( \begin{tabular}{cc|cccc} $0$ & $a$ & $0$ & $\cdots $ & $\cdots $ & $0$ \\ $-a$ & $0$ & $a$ & $0$ & $\cdots $ & $0$ \\ \hline $0$ & $a$ & & & & \\ $\vdots $ & $0$ & & $A_{t-1}$ & $B_{t-1}$ & \\ $\vdots $ & $\vdots $ & & $(-1)^{t-1}B_{t-1}$ & $A_{t-1}$ & \\ $0$ & $0$ & & & & \end{tabular} \right). \label{30} \end{equation} From the expansion formula along with the first row of (\ref{30}), it follows that \begin{equation} \mathrm{Pf}(\mathcal{M}_{2t})=a\mathrm{Pf}(\mathcal{F}_{2t-2})=af_{t-1}. \label{40} \end{equation} From (\ref{20}) and (\ref{40}), we have \begin{equation*} f_{t+1}=bg_{t}+a^{2}f_{t-1}. \end{equation*} The recurrence relation for $g_{t+1}$ can be obtained similarly. \end{proof} \begin{corollary} \label{Cor:1} $f_{n}=(-1)^{n-1}bf_{n-1}+a^{2}f_{n-2}$ with $f_{-1}=0$\ and $f_{1}=1$. \end{corollary} Corollary \ref{Cor:1} shows that the computational costs of $\mathrm{Pf}(\mathcal{F}_{n})$ and $\det (\mathcal{F}_{n}) (=\mathrm{Pf}(\mathcal{F}_{n})^2)$ are of the order $\mathcal{O}(n)$. Furthermore, the recurrences in Corollary \ref{Cor:1} have no divisions. Thus, no breakdown occurs during the computation. \section{Determinant of the skew-centrosymmetric matrix} In this section, we consider the determinant of the matrix $\mathcal{F}_{n}$ with $n=2k$. It is well known from \cite{3} that the determinant of the 2-by-2 block matrix holds \begin{equation*} \left\vert \begin{array}{cc} A & B \\ C & D \end{array} \right\vert =\det (AD-CB) \end{equation*} if $AC=CA$. Applying the above formula to $\mathcal{F}_{n}$ in Definition \ref{def:F}, the determinant of matrix $\mathcal{F}_{n}$ equals that of $\mathcal{T}_{k}:=A_k^2-(-1)^{k}B_k^2$. Thus, we have \begin{equation*} \left\vert \mathcal{F}_{n}\right\vert =\left\vert \mathcal{T}_{k}\right\vert =\det \left( \begin{array}{ccccc} -a^{2}+b^{2} & 0 & a^{2} & & \\ 0 & -2a^{2}+b^{2} & 0 & \ddots & \\ a^{2} & 0 & \ddots & \ddots & a^{2} \\ & \ddots & \ddots & -2a^{2}+b^{2} & 0 \\ & & a^{2} & 0 & -a^{2}+b^{2} \end{array} \right) _{k\times k}. \end{equation*} The matrix $\mathcal{T}_{k}$ belongs to the set of $k$-tridiagonal matrices. Sogabe and El-Mikkawy \cite{10} considered a fast block diagonalization of $k$-tridiagonal matrices using permutation matrices. Exploiting the block diagonalization method, we can rearrange the matrix $\mathcal{T}_{k}$ as below. \begin{enumerate} \item[(i)] We consider the case where $k$ is odd. Let us define the following matrices: \begin{equation*} H_{\frac{k-1}{2}}=(h_{i,j})= \begin{cases} -2a^{2}+b^{2},& \text{if $i=j$;} \\ a^{2}, & \text{if $i=j+1$ or $j=i+1$}; \\ 0, & \text{otherwise} \end{cases} \end{equation*} and \begin{equation*} K_{\frac{k+1}{2}}=(k_{i,j})= \begin{cases} -a^{2}+b^{2}, & \text{if $i=j=1$ or $i=j=\frac{k+1}{2}$;} \\ -2a^{2}+b^{2},& \text{if $i=j=2 \dots \frac{k-1}{2}$}; \\ a^{2}, & \text{if $i=j+1$ or $j=i+1$;} \\ 0, & \text{otherwise.} \end{cases} \end{equation*} Then, \begin{equation*} P^{T}\mathcal{T}_{k}P=\left( \begin{array}{c|c} H_{\frac{k-1}{2}} & 0 \\ \hline 0 & K_{\frac{k+1}{2}} \end{array} \right), \text{ } \end{equation*} where the permutation matrix $P$ is determined by using the method in \cite{10}. Obviously, \begin{equation*} \det (P^{T}\mathcal{T}_{k}P)=\det \mathcal{T}_{k}=\det \mathcal{F}_{n}=\det (H_{\frac{k-1}{2}})\det (K_{\frac{k+1}{2}}). \end{equation*} \item[(ii)] We consider the case where $k$ is even. Let us define \begin{equation*} N_{\frac{k}{2}}=(n_{i,j})= \begin{cases} -a^{2}+b^{2}, &\text{if $i=j=\frac{k}{2}$;} \\ -2a^{2}+b^{2}, &\text{if $i=j=1 \dots \frac{k}{2}-1$;} \\ a^{2}, &\text{if $i=j+1$ or $j=i+1$;} \\ 0, &\text{otherwise} \end{cases} \end{equation*} and \begin{equation*} Q_{\frac{k}{2}}=(q_{i,j})= \begin{cases} -a^{2}+b^{2}, & \text{if $i=j=1$;} \\ -2a^{2}+b^{2}, & \text{if $i=j=2\dots\frac{k}{2}$;} \\ a^{2}, & \text{if $i=j+1$ or $j=i+1$;} \\ 0, & \text{otherwise.} \end{cases} \end{equation*} Then, \begin{equation*} P^{T}\mathcal{T}_{k}P=\left( \begin{array}{c|c} N_{\frac{k}{2}} & 0 \\ \hline 0 & Q_{\frac{k}{2}} \end{array} \right). \end{equation*} Obviously, \begin{equation*} \det (P^{T}\mathcal{T}_{k}P)=\det \mathcal{T}_{k}=\det \mathcal{F}_{n}=\det (N_{\frac{k}{2}})\det (Q_{\frac{k}{2}}). \end{equation*} It can be seen that $\det (N_{\frac{k}{2}})=\det (Q_{\frac{k}{2}})$. \end{enumerate} El-Mikkawy \cite{11} obtained two-term recurrence relation for the determinants of tridiagonal matrices, i.e., \begin{equation*} v_{i}=\left\vert \begin{array}{ccccc} d_{1} & a_{1} & 0 & \ldots & 0 \\ b_{2} & d_{2} & a_{2} & \ddots & \vdots \\ 0 & b_{3} & d_{3} & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & a_{i-1} \\ 0 & \ldots & 0 & b_{i} & d_{i} \end{array} \right\vert, \end{equation*} where $v_{i}=d_{i}v_{i-1}-b_{i}a_{i-1}v_{i-2}$ for $v_{0}=1$ and $v_{-1}=0$. Using this relation and the Laplace expansion, we obtain the result. If $k$ is even, then \begin{equation*} \det (N_{\frac{k}{2}})=\det (Q_{\frac{k}{2}})=(-a^{2}+b^{2})w_{\frac{k}{2} -1}-a^{4}w_{\frac{k}{2}-2}. \end{equation*} If $k$ is odd, then \begin{align*} &\det (K_{\frac{k+1}{2}})=\left( -a^{2}+b^{2}\right) ^{2}w_{\frac{k-3}{2} }-2a^{4}(-a^{2}+b^{2})w_{\frac{k-5}{2}}+a^{8}w_{\frac{k-7}{2}}, \\ &\det (H_{\frac{k-1}{2}})=w_{\frac{k-1}{2}}, \end{align*} where \begin{equation*} w_{i}=\left\vert \begin{array}{cccc} -2a^{2}+b^{2} & a^{2} & \ldots & 0 \\ a^{2} & -2a^{2}+b^{2} & \ddots & \vdots \\ \vdots & \ddots & \ddots & a^{2} \\ 0 & \ldots & a^{2} & -2a^{2}+b^{2} \end{array} \right\vert. \end{equation*} Here $w_{i}=(-2a^{2}+b^{2})w_{i-1}-a^{4}w_{i-2}$ for $w_{0}=1$ and $w_{-1}=0$ . Consequently, for $n=2k$, we obtain \begin{enumerate} \item[(i)] If $k$ is odd, \begin{align*} \det \mathcal{F}_{n} &= \det \mathcal{T}_{k} \\ &=w_{\frac{k-1}{2}}\left( \left( -a^{2}+b^{2}\right) ^{2}w_{\frac{k-3}{2}}-2a^{4}(-a^{2}+b^{2})w_{\frac{k-5}{2 }}+a^{8}w_{\frac{k-7}{2}}\right). \end{align*} \item[(ii)] If $k$ is even, $\det \mathcal{F}_{n}=\det \mathcal{T} _{k}=\left( (-a^{2}+b^{2})w_{\frac{k}{2}-1}-a^{4}w_{\frac{k}{2}-2}\right) ^{2}. $ \end{enumerate} \section{Examples} Some examples of the Pfaffian and the determinant of the matrix $\mathcal{F}_{n}~$($n=2k$) are shown in the following tables. Here $F_{n}$, $P_{n}$, and $J_{n}$ are the $n$th Fibonacci, Pell, and Jacobsthal numbers, respectively. \begin{table}[H] \label{tPfaffian}\centering \begin{equation*} \begin{tabular}{c|l|l|l} & $a=i,b=1$ & \ $a=i,b=2$ & $a=i\sqrt{2},b=1$ \\ \cline{2-4} $k$ & \hphantom{-}\hphantom{-} $\text{Pf}{\scriptstyle (}\mathcal{F}_{2k}{\scriptstyle )}$ & \hphantom{-}\hphantom{-} $\text{Pf}{\scriptstyle (}\mathcal{F}_{2k}{\scriptstyle )}$ & \hphantom{-}\hphantom{-} $\text{Pf}{\scriptstyle (}\mathcal{F}_{2k}{\scriptstyle )}$ \\ \hline\hline \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 1$} & $\hphantom{-} F_{2}=1$ & $ \hphantom{-} P_{2}=2$ & $\hphantom{-} J_{2}=1 $ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 2$} & $-F_{3}=-2$ & $-P_{3}=-5$ & $ -J_{3}=-3$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 3$} & $-F_{4}=-3$ & $-P_{4}=-12$ & $ -J_{4}=-5$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 4$} & $\hphantom{-} F_{5}=5$ & $\hphantom{-} P_{5}=29$ & $\hphantom{-} J_{5}=11$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 5$} & $\hphantom{-} F_{6}=8$ & $\hphantom{-} P_{6}=70$ & $\hphantom{-} J_{6}=21$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 6$} & $-F_{7}=-13$ & $-P_{7}=-169$ & $-J_{7}=-43$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 7$} & $-F_{8}=-21$ & $-P_{8}=-408$ & $-J_{8}=-85$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ 8$} & $\hphantom{-} F_{9}=34$ & $\hphantom{-} P_{9}=985$ & $\hphantom{-} J_{9}=171$ \\ \multicolumn{1}{l|}{$\ \ \ \ \ \ \ \ \vdots$} & \hphantom{-}\hphantom{-}\hphantom{-}\hphantom{-} $\vdots $ & \hphantom{-}\hphantom{-}\hphantom{-}\hphantom{-} $\vdots $ & \hphantom{-} \hphantom{-}\hphantom{-}\hphantom{-} $\vdots $ \\ \hline $\equiv 0,1$ (mod $4$) & $\hphantom{-} F_{k+1}$ & $\hphantom{-} P_{k+1}$ & $\hphantom{-} J_{k+1}$ \\ $\equiv 2,3$ (mod $4$) & $-F_{k+1}$ & $-P_{k+1}$ & $-J_{k+1}$ \end{tabular} \end{equation*} \centerline{Examples of the Pfaffians} \label{tbl:determinant} \end{table} \vfill\eject \begin{table}[H] \centering \begin{equation*} \begin{tabular}{c|c|c|c} & $a=i,b=1$ & $a=i,b=2$ & $a=i\sqrt{2},b=1$ \\ \cline{2-4} $k$ & $\det { (}\mathcal{F}_{2k}{ )}$ & $\det { (}\mathcal{ F}_{2k}{ )}$ & $\det { (}\mathcal{F}_{2k}{ )}$ \\ \hline\hline \multicolumn{1}{c|}{${ 1}$} & ${ F}_{2}^{2}$ & ${ P}_{2}^{2}$ & ${ J}_{2}^{2}$ \\ \multicolumn{1}{c|}{${ 2}$} & ${ F}_{3}^{2}$ & ${ P}_{3}^{2}$ & ${ J}_{3}^{2}$ \\ \multicolumn{1}{c|}{${ 3}$} & ${ F}_{4}^{2}$ & ${ P}_{4}^{2}$ & ${ J}_{4}^{2}$ \\ \multicolumn{1}{c|}{${ 4}$} & ${ F}_{5}^{2}$ & ${ P}_{5}^{2}$ & ${ J}_{5}^{2}$ \\ \multicolumn{1}{c|}{${ 5}$} & ${ F}_{6}^{2}$ & ${ P}_{6}^{2}$ & ${ J}_{6}^{2}$ \\ \multicolumn{1}{c|}{${ 6}$} & ${ F}_{7}^{2}$ & ${ P}_{7}^{2}$ & ${ J}_{7}^{2}$ \\ \multicolumn{1}{c|}{${ 7}$} & ${ F}_{8}^{2}$ & ${ P}_{8}^{2}$ & ${ J}_{8}^{2}$ \\ \multicolumn{1}{c|}{${ 8}$} & ${ F}_{9}^{2}$ & ${ P}_{9}^{2}$ & ${ J}_{9}^{2}$ \\ \multicolumn{1}{c|}{${ \vdots}$} & ${ \vdots }$ & ${ \vdots }$ & ${ \vdots }$ \\ \multicolumn{1}{c|}{${ t}$} & ${ F}_{t+1}^{2}$ & ${ P}_{t+1}^{2}$ & ${ J}_{t+1}^{2}$ \end{tabular} \end{equation*} {Examples of the determinants} \end{table} \section{Acknowledgment} This work has been supported in part by JSPS KAKENHI (Grant No.~26286088). The authors sincerely appreciate the referee's comments that enhanced the quality of the manuscript. \begin{thebibliography}{9} \bibitem{1} A. L. Andrew, Centrosymmetric matrices, {\it SIAM Rev.} {\bf 40} (1988), 697--698. \bibitem{2} A. L. Andrew, Eigenvectors of certain matrices, {\it Lin. Alg. Appl.} {\bf 7} (1973), 151--162. \bibitem{3} F. Zhang, {\it Matrix Theory: Basic Results and Techniques}, Springer, 1999. \bibitem{4} L. Datta and S. Morgera, On the reducibility of centrosymmetric matrices- applications in engineering problems, {\it Circ. Syst. Signal. Pr.} {\bf 8} (1989), 71--96. \bibitem{5} M. El-Mikkawy and F. Atlan, On solving centrosymmetric linear systems, {\it Appl. Math.} {\bf 4} (2013), 21--32. \bibitem{6} J. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, {\it Amer. Math. Monthly} {\bf 92} (1985), 711--717. \bibitem{7} R. Vein and P. Dale, {\it Determinants and Their Applications in Mathematical Physics}, Springer, 1999. \bibitem{10} T. Sogabe and M. El-Mikkawy, Fast block diagonalization of $k$-tridiagonal matrices, {\it Appl. Math. Comput.} {\bf 218} (2011), 2740--2743. \bibitem{11} M. El-Mikkawy, A note on a three-term recurrence for a tridiagonal matrix, {\it Appl. Math. Comput.} {\bf 139} (2003), 503--511. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 15A15; Secondary 15A23. \noindent \emph{Keywords: } Pfaffian, determinant, skew-centrosymmetric matrix. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum {A000045}, \seqnum {A000129}, \seqnum{A001045}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received June 20 2016; revised versions received July 8 2016; January 18 2017; February 2 2017. Published in {\it Journal of Integer Sequences}, February 11 2017. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document} .