Conjugate Transpose of a Matrix #
Definition:
Let \(\pmb{A\in M_{m\times n} (F)}\) be a \(\pmb{m\times n}\) matrix. Then the \( \fcolorbox{blue}{white}{\pmb{Conjugate Transpose}}\) or \( \fcolorbox{blue}{white}{\pmb{Adjoint}}\) of \(\pmb{A}\) is denoted by \(\pmb{A^{*}}\) and is defined by the \(\pmb{n\times m}\) matrix such that
\(\pmb{A^{*}_{ij}=\overline{A_{ji}} }\) #
Example-1:
Let \(\pmb{A=\begin{bmatrix}
2+3i & -3+2i \\
7 & 5i
\end{bmatrix} }\)
then \(\pmb{\overline{A}=\begin{bmatrix}
2-3i & -3-2i \\
7 & -5i
\end{bmatrix} }\)
Therefore the conjugate transpose of \(\pmb{A}\) ,
$$\pmb{A^{*} =\begin{bmatrix}
2-3i & 7\\
-3-2i & -5i
\end{bmatrix} }$$
Conclusion:
If \(\pmb{A\in M_{m\times n} (R)}\) then \(\pmb{A^{*}=A^{t}}\) #
Proof:
Let \(\pmb{A\in M_{m\times n} (R)}\) then \(\pmb{A=[A_{ij}]_{m\times n}}\) where each \(\pmb{A_{ij}\in \mathcal{R} }\)
Now
\begin{align*}
\pmb{A^{*}_{ij}} & = \pmb{\overline{A_{ji}} }\\
& = \pmb{A_{ji}~since~\overline{A_{ji}}=A_{ji} }\\
\end{align*}
Therefore, \( \fcolorbox{blue}{white}{\( \pmb{ A^{*}=A^{t} } \)} \).