Transpose and adjoint

Transpose: The transpose of an $m \times n$ matrix $A$, denoted $A^T$, is the $n \times m$ matrix obtained by flipping rows and columns:

$$(A^T)_{ij} = a_{ji}$$

Example:
$$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$$

Properties:

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(AB)^T = B^T A^T$
• $(A^{-1})^T = (A^T)^{-1}$

Adjoint (conjugate transpose): For a complex matrix $A$, the adjoint $A^*$ (or $A^H$) is the conjugate of the transpose:

$$A^* = (\overline{A})^T$$

For real matrices, $A^* = A^T$. A matrix is Hermitian if $A^* = A$, and symmetric if $A^T = A$.