| Version 2 |
Version 1 |
| \def\dtra{\hspace{0.04cm} ^{\mbox{\scriptsize{T}}} \hspace{0.02cm}} |
|
| \def\htra{\hspace{0.04cm} ^{\mbox{\scriptsize{H}}} \hspace{0.02cm}} |
|
| {\bf Definition} If $A$ is a complex matrix, then the |
{\bf Definition} If $A$ is a complex matrix, then the |
| \emph{conjugate transpose} $A^\ast$ is the matrix |
\emph{conjugate transpose} $A^\ast$ is the matrix |
| $A^\ast = \bar{A}\dtra$, where $\bar{A}$ is |
$A^\ast = \bar{A}\dtra$, where $\bar{A}$ is |
| the complex conjugate of $A$, and $A\dtra$ is the |
the complex conjugate of $A$, and $A\dtra$ is the |
| transpose of $A$. |
transpose of $A$. |
| It is clear that for real matrices, the conjugate transpose coincides with |
It is clear that for real matrices, the conjugate transpose coincides with |
| the transpose. |
the transpose. |
| The conjugate transpose of $A$ is also called the \emph{adjoint matrix} of $A$, |
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 \emph{Hermitian conjugate} of $A$ (whence one usually writes $A^\ast = A\htra$). |
| The notation $A^\dagger$ is also used for the conjugate transpose. In \cite{eves}, |
The notation $A^\dagger$ is also used for the conjugate transpose. In \cite{eves}, |
| $A^\ast$ is also called the \emph{tranjugate} of $A$. |
$A^\ast$ is also called the \emph{tranjugate} of $A$. |
| An overview of some of these namings and notations is given in \cite{ew_adjointmatrix}. |
An overview of some of these namings and notations is given in \cite{ew_adjointmatrix}. |
| \subsubsection{Properties} |
\subsubsection{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $A$ and $B$ are complex matrices of same dimension, and $\alpha,\beta$ |
\item If $A$ and $B$ are complex matrices of same dimension, and $\alpha,\beta$ |
| are complex constants, then |
are complex constants, then |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| (\alpha A + \beta B)^\ast &=& \overline{\alpha} A^\ast + \overline{\beta} B^\ast,\\ |
(\alpha A + \beta B)^\ast &=& \overline{\alpha} A^\ast + \overline{\beta} B^\ast,\\ |
| A^{\ast\ast} &=& A. |
A^{\ast\ast} &=& A. |
| \end{eqnarray*} |
\end{eqnarray*} |
| \item If $A$ and $B$ are complex matrices such that $AB$ is defined, then |
\item If $A$ and $B$ are complex matrices such that $AB$ is defined, then |
| $$ (AB)^\ast = B^\ast A^\ast.$$ |
$$ (AB)^\ast = B^\ast A^\ast.$$ |
| \item If $A$ is a complex square matrix, then |
\item If $A$ is a complex square matrix, then |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \det (A^\ast) &=& \overline{ \det{A}}, \\ |
\det (A^\ast) &=& \overline{ \det{A}}, \\ |
| \operatorname{trace}(A^\ast) &=& \overline{ \operatorname{trace}{A}}, \\ |
\operatorname{trace}(A^\ast) &=& \overline{ \operatorname{trace}{A}}, \\ |
| (A^\ast)^{-1} &=& (A^{-1})^\ast, |
(A^\ast)^{-1} &=& (A^{-1})^\ast, |
| \end{eqnarray*} |
\end{eqnarray*} |
| where $\operatorname{trace}$ and $\operatorname{det}$ are the trace |
where $\operatorname{trace}$ and $\operatorname{det}$ are the trace |
| and the determinant operators, and $^{-1}$ is the inverse operator. |
and the determinant operators, and $^{-1}$ is the inverse operator. |
| \item Suppose $<\cdot, \cdot>$ is the standard inner product on $\sC^n$. |
\item Suppose $<\cdot, \cdot>$ is the standard inner product on $\sC^n$. |
| Then for an arbitrary complex $n\times n$ matrix $A$, |
Then for an arbitrary complex $n\times n$ matrix $A$, |
| and vectors $x,y\in \sC^n$, we have that |
and vectors $x,y\in \sC^n$, we have that |
| $$ <Ax,y> = <x,A^\ast y>.$$ |
$$ <Ax,y> = <x,A^\ast y>.$$ |
| \end{enumerate} |
\end{enumerate} |
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {eves} H. Eves, |
\bibitem {eves} H. Eves, |
| \emph{Elementary Matrix Theory}, |
\emph{Elementary Matrix Theory}, |
| Dover publications, 1980. |
Dover publications, 1980. |
| \bibitem{ew_adjointmatrix} E. Weisstein, Eric W. Weisstein's world of mathematics, |
\bibitem{ew_adjointmatrix} E. Weisstein, Eric W. Weisstein's world of mathematics, |
| \PMlinkexternal{entry on Adjoint Matrix}{http://mathworld.wolfram.com/AdjointMatrix.html}. |
\PMlinkexternal{entry on Adjoint Matrix}{http://mathworld.wolfram.com |
| % \bibitem{wiki_conjtranspose} Wikipedia, |
0 & 0 & 0 & \cdots & 0 |
| % \PMlinkexternal{conjugate transpose}{http://www.wikipedia.org/wiki/Conjugate_transpose} |
\end{bmatrix} $$ |
| \end{thebibliography} |
A strictly lower triangular matrix is of the form |
| \subsubsection{See also} |
$$ \begin{bmatrix} |
| \begin{itemize} |
0 & 0 & 0 & \cdots & 0 \\ |
| \item Wikipedia, |
a_{21} & 0 & 0 & \cdots & 0 \\ |
| \PMlinkexternal{conjugate transpose}{http://www.wikipedia.org/wiki/Conjugate_transpose} |
a_{31} & a_{32} & 0 & \cdots & 0 \\ |
| \end{itemize} |
\vdots & \vdots & \vdots & \ddots & \vdots \\ |
|
a_{n1} & a_{n2} & a_{n3} & \cdots & 0 |
|
\end{bmatrix} $$ |
