15-00-ConjugateTranspose.tex

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\pmtitle{conjugate transpose}
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\pmtype{Definition}
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\pmsynonym{adjoint matrix}{ConjugateTranspose}
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\pmrelated{Transpose}

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{\bf Definition} If $A$ is a complex matrix, then the 
\emph{conjugate transpose} $A^\ast$ is the matrix 
$A^\ast = \bar{A}\dtra$, where $\bar{A}$ is
the complex conjugate of $A$, and $A\dtra$ is the 
transpose of $A$. 

It is clear that for real matrices, the conjugate transpose coincides with
the transpose. 

\subsubsection{Properties}
\begin{enumerate}
\item If $A$ and $B$ are complex matrices of same size, and $\alpha,\beta$
are complex constants, then
\begin{eqnarray*}
  (\alpha A + \beta B)^\ast &=& \overline{\alpha} A^\ast + \overline{\beta} B^\ast,\\
  A^{\ast\ast} &=& A.
\end{eqnarray*}

\item If $A$ and $B$ are complex matrices such that $AB$ is defined, then 
$$ (AB)^\ast = B^\ast A^\ast.$$
\item If $A$ is a complex square matrix, then 
\begin{eqnarray*}
 \det (A^\ast) &=& \overline{ \det{A}}, \\
\operatorname{trace}(A^\ast) &=& \overline{ \operatorname{trace}{A}}, \\
(A^\ast)^{-1} &=& (A^{-1})^\ast,
\end{eqnarray*}
where $\operatorname{trace}$ and $\operatorname{det}$ are the trace 
and the determinant operators, and $^{-1}$ is the inverse operator. 
\item Suppose $\langle \cdot, \cdot \rangle$ is the standard inner product on $\sC^n$. 
Then for an arbitrary complex $n\times n$ matrix $A$,  
and vectors $x,y\in \sC^n$, we have 
$$ \langle Ax,y\rangle = \langle x,A^\ast y \rangle.$$
\end{enumerate}

\subsubsection*{Notes}
The conjugate transpose of $A$ is also called the \emph{adjoint matrix} of $A$, 
the \emph{Hermitian conjugate} of $A$ (whence one usually writes $A^\ast = A\htra$).
The notation $A^\dagger$ is also used for the conjugate transpose \cite{pease}. 
In \cite{eves}, $A^\ast$ is also called the \emph{tranjugate} of $A$.


\begin{thebibliography}{9}
 \bibitem {eves} H. Eves, \emph{Elementary Matrix Theory}, Dover publications, 1980.
\bibitem {pease} M. C. Pease,
 \emph{Methods of Matrix Algebra},  Academic Press, 1965.
 \end{thebibliography}


\subsubsection*{See also}
\begin{itemize}
 \item Wikipedia, 
 \PMlinkexternal{conjugate transpose}{http://www.wikipedia.org/wiki/Conjugate_transpose}
\end{itemize}
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