Definitions
The transpose of a matrix $A$ is $A\T$ with entries $$A\T_{ij} = A_{ji}$$ and the conjugate transpose of $A$ is $A^*$ (or $A^\textrm{H}$) with entries $$A^*_{ij} = \baa{A_{ji}}$$
We have $A^* = \baa{A\T}$. And when $A$ is real, $A^* = A\T$.
Special Matrices
Let $A$ be a square matrix. The following properties have fancy names.
- Symmetric $$A\T = A$$
- Hermitian $$A\T = A^*$$
- Skew-Symmetric $$A\T = -A$$
- Skew-Hermitian $$A\T = -A^*$$
- Orthogonal (although the columns are orthonormal) $$A\T = A^{-1}$$
- Unitary $$A^* = A^{-1}$$
- Normal ($\supset$ Unitary) $$A^*A = AA^*$$