The Hermitian adjoint of a matrix is the same as its transpose except that along with switching row and column elements you also complex conjugate all the elements. If all the elements of a matrix are real, its Hermitian adjoint and transpose are the same. In terms of components,

$$\left(A_{ij}\right)^\dagger=A_{ji}^*\text{.}\tag{3.5.1}$$

For example, if

$$A = \left(\begin{array}{c} 1\\ i\\ -2i \end{array} \right)\tag{3.5.2}$$

then

$$A^\dagger = \left(\begin{array}{ccc} 1\amp -i\amp 2i \end{array} \right)\text{.}\tag{3.5.3}$$

A matrix is called Hermitian if it is equal to its adjoint, $A = A^\dagger\text{.}$

Try it for yourself by computing $B^\dagger$ if

$$B=\left(\begin{array}{cc} 1\amp i\\ -5i\amp i \end{array} \right)\text{.}\tag{3.5.4}$$