conjugate transposeConjugateTranspose2003-06-21 13:49:422006-09-13 22:14:20Definition% this is the default PlanetMath preamble. as your knowledge
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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}