Hermitian matrix

Introduction

A Hermitian matrix (or self-adjoint matrix) is one which is equal to its Hermitian adjoint (also known as its conjugate transpose). That is to say that every entry in the tranposed matrix is replaced by its complex conjugate:
$a_{i,j}={\overline {a_{i,j}}}$,
or in matrix notation:
$\mathbf {A=A^{*}(={\overline {A'}})}$

Forming the Hermitian adjoint

To form the Hermitian adjoint of the matrix $\mathbf {A} ={\begin{pmatrix}a&b+{\mathit {i}}c&e+{\mathit {i}}f\\b-{\mathit {i}}c&d&h+{\mathit {i}}k\\e-{\mathit {i}}f&h-{\mathit {i}}k&g\end{pmatrix}}$:

1. Form the transpose matrix $\mathbf {A'} ={\begin{pmatrix}a&b-{\mathit {i}}c&e-{\mathit {i}}f\\b+{\mathit {i}}c&d&h-{\mathit {i}}k\\e+{\mathit {i}}f&h+{\mathit {i}}k&g\end{pmatrix}}$, by replacing $a_{i,j}$ with $a_{j,i}$.
2. Take the complex conjugate of each entry to form the Hermitian adjoint:

$\mathbf {A^{*}={\overline {A'}}=} {\begin{pmatrix}a&b+{\mathit {i}}c&e+{\mathit {i}}f\\b-{\mathit {i}}c&d&h+{\mathit {i}}k\\e-{\mathit {i}}f&h-{\mathit {i}}k&g\end{pmatrix}}$.

We find that $\mathbf {A=A^{*}}$.

Properties

Entries on the main diagonal

It may be seen that all entries on the main diagonal of a Hermitian matrix must be real.
i.e. $a_{i,i}\in \Re$

References

Matrices and Determinants, 9th edition by A.C Aitken