How can I construct a $n\times n$ complex hermitian matrix?

Question

As indicated in the title, I need to build a complex hermitian matrix of $ n \times n $ numerically, but I am somewhat lost and I don't know how to do it. Anyone have any ideas to build it?

Answer 1

The requirement for a matrix $A$ to be Hermitian is that, for each element $a_{ij}$, it must be the case that $a_{ij}=a_{ji}^*$, where the $*$ represents the complex conjugate.

Therefore, all you need to do to make a complex Hermitian matrix is to pick any complex values for elements $a_{ij}$ where $1\leq i\leq n$ and $i\leq j\leq n$ (i.e. values for the upper right section of the matrix) and then fill in the rest of the matrix such that $a_{ij}=a_{ji}^*$ holds true. Note that along the diagonal $i=j$, so the elements $a_{ii}$ must be real in order for $a_{ij}=a_{ji}^*$ to be true for diagonal entries.

An example of a $4\times4$ matrix:

First, fill in the upper diagonal with any complex values, and the diagonal with any real values.

$$ \left(\begin{matrix} 1 & i & -i & 2+i\\ \, & 3 & -4+2i & -1-i\\ \, & \, & -2 & 0.5-3i\\ \, & \, & \, & \pi \\ \end{matrix}\right) $$

Then just fill in the rest with the complex conjugates

$$ \left(\begin{matrix} 1 & i & -i & 2+i\\ -i & 3 & -4+2i & -1-i\\ i & -4-2i & -2 & 0.5-3i\\ 2-i & -1+i & 0.5+3i & \pi \\ \end{matrix}\right) $$