The Conjugate Transpose of a Matrix

We are about to look at an important theorem which will give us a relationship between a matrix that represents the linear transformation $T$ and a matrix that represents the adjoint of $T$, $T^*$. Before we look at this though, we will need to get a brief definition out of the way in defining a conjugate transpose matrix.

Definition: If $A$ is an $m \times n$ matrix with entries from the field $\mathbb{F}$, then the Conjugate Transpose of $A$ is obtained by taking the complex conjugate of each entry in $A$ and then transposing $A$.

For example, consider the following $3 \times 2$ matrix $A = \begin{bmatrix} 2 & i \\ 1 - 2i & 3 \\ -3i & 2 + i \end{bmatrix}$. Then the conjugate transpose of $A$ is obtained by first taking the complex conjugate of each entry to get $\begin{bmatrix} 2 & -i \\ 1 + 2i & 3 \\ 3i & 2 - i \end{bmatrix}$, and then transposing this matrix to get:

(1)

\begin{bmatrix} 2 & 1 + 2i & 3i \\ -i & 3 & 2 - i \end{bmatrix}