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

 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:

 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}{\( \pmb{ A^{*}=A^{t} } \)} \).