Conjugate Transpose Matrix

Topics: Algebra - Matrix

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(definition)

Let $A\in M_{m\times n}(\mathbb{K})$. Let $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$.

We define the conjugate transpose matrix of $A$, denoted as $A^{\ast }$, with a size of $n\times m$, as the conjugate of its transpose:

$$A^{\ast }:=\left(\begin{array}{cc}\overline{a}& \overline{c}\\ \overline{b}& \overline{d}\end{array}\right)$$

Example

Let $A=\left(\begin{array}{cc}i& 1+2i\\ 2& 3+4i\end{array}\right)$, then:

$$A^{\ast }=\left(\begin{array}{cc}-i& 2\\ 1-2i& 3-4i\end{array}\right)$$