Conjugate transpose

Definition

Let $A = (a_{ij})$ be an $m \times n$ matrix with complex entries. The conjugate transpose of $A$ is the $n \times m$ matrix $A^*$ with entries $$(A^*)_{ij} = \overline{a_{ji}}$$ for $1 \le i \le n$ and $1 \le j \le m$, where $\overline{z}$ is the complex conjugate of $z$.

Remarks

Other notations are $A^H$ and $A^\dagger$. For a real matrix, the conjugate transpose is the transpose. The rules $(A^*)^* = A$ and $(AB)^* = B^* A^*$ hold. For the standard inner product on $\C^n$, $\langle Ax, y \rangle = \langle x, A^* y \rangle$.

Explanations

L1 human

The conjugate transpose flips a complex matrix over its diagonal and conjugates each entry. For example, the conjugate transpose of $\begin{pmatrix} 1 & i \\ 0 & 2 \end{pmatrix}$ is $\begin{pmatrix} 1 & 0 \\ -i & 2 \end{pmatrix}$. For complex matrices it plays the role that the transpose plays for real matrices.