Conjugate Transpose

linear-algebra

Definition

Conjugate Transpose

Let $A\in F^{m\times n}$ be a matrix over a field $F$.

The conjugate transpose of $A$, written $A^{†}$ or $A^{\ast }$, is the matrix obtained by first taking the complex conjugate of each element and then taking the matrix transpose of that conjugated matrix, i.e.:

$$A^{†}=(\overline{A}{)}^{\top }$$

Einstein Notation: $(A^{†}{)}_{ij}=\overline{A_{ji}}$