Hermitian Matrix Diagonal

As we know a Hermitian matrix $A$ satisfies the property $\overline{A^T} = A$. Let’s consider the diagonal elements of $A$ denoted by $a_{ii}$.

According to the definition of the conjugate transpose, the diagonal elements of $\overline{A^T}$ are given by $\overline{a_{ii}^T} = a_{ii}$. Since $\overline{A^T} = A$, we have $a_{ii} = \overline{a_{ii}}$.

So considering $a_{ii} = x + iy$, we have that $x + iy = x - iy$ which implies that $y = -y$ and $y = 0$. Therefore, the diagonal elements of a Hermitian matrix are real.