{{Comments on the Matrix Representations of Quaternions}}
{\Large Gisele Ducati}
{\em Departamento de Matem\'atica, UFPR}
{CP 19081, 81531-970 Curitiba (PR) Brasil}
math0310319.tex


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\def\0{\mbox{\tiny $0$}}
\def\1{\mbox{\tiny $1$}}
\def\2{\mbox{\tiny $2$}}
\def\3{\mbox{\tiny $3$}}
\def\4{\mbox{\tiny $4$}}
\def\5{\mbox{\tiny $5$}}
\def\6{\mbox{\tiny $6$}}
\def\7{\mbox{\tiny $7$}}
\def\8{\mbox{\tiny $8$}}
\def\9{\mbox{\tiny $9$}}


\def\inv{\mbox{\tiny $-$$1$}}

\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\def\H{\mathbb{H}}
\def\X{\mathbb{X}}

\def\Ri{\mbox{\tiny $R$}}
\def\Le{\mbox{\tiny $L$}}

%\def\c{\mbox{\tiny $c$}}


\def\q{\mbox{\tiny $q$}}
\def\p{\mbox{\tiny $p$}}

\def\h{\mbox{\tiny $h$}}

\def\m{\mbox{\tiny $m$}}

\def\u{\mbox{\tiny $u$}}
\def\v{\mbox{\tiny $v$}}

\def\qm{\mbox{\tiny $-$$q$}}
\def\qa{\mbox{\tiny $a$$-$$q$}}

\def\ij2{\mbox{\tiny $\frac{i+j}{2}$}}
\def\ji2{\mbox{\tiny $\frac{j-i}{2}$}}

%\def\i{\mbox{\tiny $i$}}
\def\j{\mbox{\tiny $j$}}
\def\k{\mbox{\tiny $k$}}

\def\mi{\mbox{\tiny $-$}}
\def\pl{\mbox{\tiny $+$}}


\def\z{\mbox{\tiny $z$}}

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\begin{document}

\title{{\huge {\bf Comments on the Matrix Representations of
Quaternions}}}
%[J.~Math.~Phys. 42, 2236-2265 (2001)]
\author{
{\Large \sf  Gisele Ducati}\\
{\em Departamento de Matem\'atica, UFPR}\\
{\small  CP 19081, 81531-970  Curitiba (PR) Brasil}\\
{\small  \tt ducati@mat.ufpr.br} }

\maketitle

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%%%%%%%%%%%% DATE ABSTRACT PACS % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\date{{\em May 16, 2003}}


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%\PACS{ {02.10.Hh} \and  {02.30.Tb~~} {{\bf MSC}. 11R52} \and
%{30G35} \and {46S20}{}}

% Warning: No PACS code given

%02.10.Hh Rings and algebras
%02.10.Ud Linear algebra
%02.10.Yn Matrix theory


%02.30.Hq Ordinary differential equations
%02.30.Jr Partial differential equations
%02.30.Tb Operator theory


%03.65.-w Quantum mechanics
%03.65.Ca Formalism
%03.65.Ta Foundations of quantum mechanics; measurement theory

%\maketitle


\begin{quote}
{\bf Abstract} ~Some comments are made on the matrices which serve
as the basis of a quaternionic algebra. We show that these
matrices are related with the quaternionic action of the imaginary
units from the left and from the right.

\noindent {\bf MSC.} 11R52 - 15A33.
\end{quote}


%\titlerunning{COMMENTS ON "MATRIX REPRESENTATIONS OF QUATERNIONS"}

\maketitle


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%                               INTRODUCTION
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%\section*{{\normalsize I. INTRODUCTION}}

We start by recalling the notation of the referred
paper~\cite{FGT03}. It's well known that a quaternion
\[
q = a + b h + c j + d k~,~~~~a,b,c,d \in \mathbb{R}~,
\]
where $h, j$ and $k$ denote the imaginary units of the algebra of
quaternions
\[
h^2 = j^2 = k^2 = h j k = -1~,
\]
can be written in matrix form as
\[
Q = a I + b H + c J + d K~,
\]
with
\[
H^2 = J^2 = K^2 = H J K = - I~.
\]
Like pointed out in the paper~\cite{FGT03} the matrices $H, J$ and
$K$ can be chosen in different ways so adopting skew-symmetric
signed permutation matrices with a plus one in its first row we
find six signed permutation matrices which arrange themselves to
form two Hamiltonian systems, given by $H_1, J_1, K_1$ and $H_2,
J_2, K_2$.

We emphasize that the first system represents the left action of
the imaginary units $j, h$ and $k$, respectively, and the matrices
of the second system represents, respectively, the right action of
the quaternionic imaginary units, $h, j$ and $k$. In both cases we
should multiply the units by $- 1$. Let $q = q_0 + h q_1 + j q_2 +
k q_3$ and $p = p_0 + h p_1 + j p_2 + k p_3$ real quaternions. If
we distinguish between the left and right action of the
quaternionic imaginary units $h$, $j$, and $k$ by introducing the
operators $L_{q}$ and $R_{p}$ whose action on quaternionic
functions $\Psi$, $\Psi: \H \to \R$, is given by
\begin{equation}
L_{q} \Psi = q \, \Psi~~~\mbox{and}~~~ R_{p} \Psi =\Psi \, p~,
\end{equation}
where
\[
L_h^2 = L_j^2 = L_k^2 = R_h^2 = R_j^2 = R_k^2 = L_h L_j L_k = R_k
R_j R_h = -1~,
\]
and
\begin{equation}
\left[ \, L_{q} \, , \, R_{p} \, \right] = L_q\,R_p - R_p \, L_q =
0~,
\end{equation}
we find the following {\em real} matrix
representation~\cite{DD99,DD01}
\begin{equation}
\label{rrep} L_{q} \leftrightarrow
 \left( \begin{array}{rrrr} q_{\0} & $-$
q_{\1} & $-$
q_{\2} & $-$ q_{\3} \\
q_{\1} & q_{\0} & $-$
q_{\3} & q_{\2} \\
q_{\2} & q_{\3} &
q_{\0} & $-$ q_{\1} \\
q_{\3} & $-$ q_{\2} & q_{\1} &  q_{\0} \end{array}
\right)~~~\mbox{and}~~~R_{p} \leftrightarrow  \left(
\begin{array}{rrrr} p_{\0} & $-$ p_{\1} & $-$
p_{\2} & $-$ p_{\3} \\
p_{\1} & p_{\0} &
p_{\3} & $-$ p_{\2} \\
p_{\2} & $-$ p_{\3} &
p_{\0} &  p_{\1} \\
p_{\3} &  p_{\2} & $-$ p_{\1} &  p_{\0} \end{array} \right)~.
\end{equation}
It's interesting to note that the six matrices
\begin{equation}
\label{op} L_h,~ L_j,~ L_k,~ R_h,~ R_j~~\mbox{and}~~R_k
\end{equation}
are the generators of the one-dimensional quaternion unitary group
linear from the right. See \cite{DD99,DUC02} for further details.
In addition, it's more natural to arrange the six signed
permutation matrices cited in \cite{FGT03} as given in (\ref{op}).

Now, let us see how many triplets, obeying the quaternionic
algebra, is possible to write with these elements. We should
consider each element from (\ref{op}) and it's conjugate (or it's
negation) which give 12 elements. Obviously, fixing a imaginary
unit, the second one must be chosen in a way that it does not
commute with the previously fixed one, otherwise, the
multiplication of the imaginary units will not square to minus
one, contradicting the quaternion algebra. So, if $L_h$ has been
chosen for the first imaginary unit, the remaining possibilities
are $\pm L_j$ and $\pm L_k$ and the third imaginary unit is
determined by the multiplication of the first two. So we have $12$
possibilities for the first imaginary unit and $4$ for the second,
that means, $12 \times 4 \times 1$, totalizing 48 possibilities.
Of course, we can consider all automorphisms of the quaternion
algebra, or alternatively, thinking in a geometric way, any
quaternion can be rotated in the three dimensional space, then we
should consider the transformation
\[
q \to q_u = u q u^{-\1}~,~~~|u|=1~.
\]
Note that
\[
h_u^2 = j_u^2 = k_u^2 = h_u j_u k_u = -1~.
\]
This similarity transformation can be made with $L_h,~ L_j,~ L_k,~
R_h,~ R_j$ and $R_k$. The automorphism group of the units is the
$O(3)$ subgroup of $O(4)$ given by
\[
\Omega' = U \Omega U^{\dag}~.
\]


\begin{thebibliography}{99}

\bibitem{FGT03}
R. W. Farebrother, J. Gro\ss~ and S. Troschke, Matrix
representation of quaternions, {\it Lin. Alg. Appl.} {\bf 362},
251--255 (2003).

\bibitem{DD99}
S. De Leo and G. C. Ducati, Quaternionic groups in physics, {\it
Int. J. Theor. Phys.} {\bf 38}, 2197-2220 (1999).

\bibitem{DD01}
S. De Leo  and G. C. Ducati, Quaternionic differential operators,
{\it J.\,Math.\,Phys.} {\bf 42}, 2236--2265 (2001).

\bibitem{DUC02}
G. Ducati, {\it Operadores diferenciais quaterni\^onicos e
aplica\c{c}\~oes em F\'{\i}sica}, Ph.D. Thesis, IMECC-UNICAMP
(2002).
\end{thebibliography}



\end{document}


With these six different imaginary units we have different ways to
select triplets. Obviously, we cannot mix left and right actions.


It's well known that quaternions don't commute. So, we have to
differentiate left from the right action of a quaternion on a
quaternionic function of real variables. Denoting the left action
by $L_q$ and the right action by $R_p$, $q, p \in \H$ we have the
following possibilities
\[
\pm L_i~,~\pm L_j~,~\pm L_k~~~\mbox{and}~~~\pm R_i~,~\pm R_j~,~\pm
R_k~~~~~[L_q, R_p]=0
\]

Let us give a brief introduction to the quaternion algebra,
$\mathbb{H}$. A quaternion $q$ is usually defined as
\begin{equation}
q = q_{\0} + i q_{\1} + j q_{\2} + k q_{\3}~,~~~q_{\0,\1,\2,\3}
\in \R~,
\end{equation}
where $i, j, k$ are the imaginary units satisfying
\begin{equation}
\label{alg} i^2 = j^2 = k^2 = i j k = -1~.
\end{equation}
Quaternions are added and multiplied according to the usual law of
arithmetics, excluding the commutative law of multiplication as
can be easily seen from (\ref{alg}). Since quaternions are not
commutative, it is convenient to distinguish between

math0310319.extracted_bib
\begin{thebibliography}{99}

\bibitem{FGT03}
R. W. Farebrother, J. Gro\ss~ and S. Troschke, Matrix
representation of quaternions, {\it Lin. Alg. Appl.} {\bf 362},
251--255 (2003).

\bibitem{DD99}
S. De Leo and G. C. Ducati, Quaternionic groups in physics, {\it
Int. J. Theor. Phys.} {\bf 38}, 2197-2220 (1999).

\bibitem{DD01}
S. De Leo  and G. C. Ducati, Quaternionic differential operators,
{\it J.\,Math.\,Phys.} {\bf 42}, 2236--2265 (2001).

\bibitem{DUC02}
G. Ducati, {\it Operadores diferenciais quaterni\^onicos e
aplica\c{c}\~oes em F\'{\i}sica}, Ph.D. Thesis, IMECC-UNICAMP
(2002).
\end{thebibliography}