Skew-Hermitian Diagonal Elements Property

A matrix '$A$' is called skew-Hermitian if its conjugate transpose '$A^\dagger$' is equal to its negative, i.e., $A^\dagger = -A$.

Let the elements of the matrix '$A$' be denoted by '$a_{ij}$'. The conjugate transpose '$A^\dagger$' has elements $(A^\dagger)_{ij} = \overline{a_{ji}}$ (where $\overline{z}$ denotes the complex conjugate of $z$).

The condition for a skew-Hermitian matrix is $A^\dagger = -A$. This means $(A^\dagger)_{ij} = -a_{ij}$ for all $i$ and $j$. Specifically, for the diagonal elements where $i = j$, the condition becomes:

Now, let's represent a diagonal element '$a_{ii}$' in terms of its real part '$x$' and imaginary part '$y$':

The complex conjugate is:

Substituting these into the condition $\overline{a_{ii}} = -a_{ii}$:

Adding '$iy$' to both sides gives:

This equation implies:

Since the real part '$x$' must be zero, the diagonal element '$a_{ii}$' must be of the form $0 + iy$, which simplifies to '$iy$'.

• If $y = 0$, then $a_{ii} = i(0) = 0$.
• If $y \neq 0$, then $a_{ii}$ is a non-zero purely imaginary number.

Therefore, the diagonal elements of a skew-Hermitian matrix must be purely imaginary or zero.