Conjugate Transpose

Definition

• With $A:=(a_{j,k})\in {\mathbb{C}}^{n\times n}$
• Then $A^{\ast }:=(\overline{a_{k,j}})$ is the Conjugate Transpose of $A$

Computing Process

Ensure that you work with an Orthonormal Basis.

1. Complex Conjugate all entries
2. Matrix Transposition

Example

2 & 1 + i\\ i & 3 - i \end{array}\right]$$ We find $A^{*}$

A^{*} = \left[\begin{array}{cc} 2 & -i\ 1-i & 3+i\ \end{array}\right]

# Theorems - [[Conjugate Transpose of Itself is the Original Matrix]]