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 transposed matrix is replaced by its complex conjugate:
$a_{i,j}={\overline {a_{j,i}}}$,
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. $a_{i,i}\in \Re$

Real-valued Hermitian matrices

A real-valued Hermitian matrix is a real symmetric matrix and hence the theorems of the latter are special cases of theorems of the former.

Addition

The sum of two Hermitian matrices is Hermitian. Any square matrix, $\mathbf {C}$, can be written as the sum of a Hermitian matrix, $\mathbf {A}$, and skew-Hermitian matrix, $\mathbf {B}$:

$\mathbf {C=A+B}$ where $\mathbf {A} ={\frac {1}{2}}(\mathbf {C+C^{*}} )$ and $\mathbf {B} ={\frac {1}{2}}(\mathbf {C-C^{*}} )$

Normal

All Hermitian matrices are normal, i.e. $\mathbf {AA^{*}=A^{*}A}$, and thus the finite dimensional spectral theorum applies. This means that any Hermitian matrix can be diagonalised by a unitary matrix, all its entries have real values

Eigenvalues

All the eigenvalues of Hermitian matrices are real. Eigenvectors with distinct eigenvalues are orthogonal.

Pauli spin matrices

Any 2x2 Hermitian matrix may be written as a linear combination of the Pauli spin matrices. These matrices have unrelated uses in quantum mechanics, but may be used usefully for this purpose. They thus form an orthogonal basis for the real Hilbert space of 2x2 Hermitian matrices.

$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\sigma _{y}={\begin{pmatrix}0&-{\mathit {i}}\\{\mathit {i}}&0\end{pmatrix}},\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

$\mathbf {A} =a\mathbf {I} +b\sigma _{x}+c\sigma _{y}+d\sigma _{z}={\begin{pmatrix}a&0\\0&a\end{pmatrix}}+{\begin{pmatrix}0&b\\b&0\end{pmatrix}}+{\begin{pmatrix}0&-{\mathit {i}}c\\{\mathit {i}}c&0\end{pmatrix}}+{\begin{pmatrix}d&0\\0&-d\end{pmatrix}}={\begin{pmatrix}a+d&b-{\mathit {i}}c\\b+{\mathit {i}}c&a-d\end{pmatrix}}$

Skew-Hermitian Matrices

A skew-Hermitian matrix is one which is equal to the negative of its Hermitian adjoint:
$\mathbf {A=-A^{*}}$

$a_{i,j}=-{\overline {a_{j,i}}}$

e.g. $\mathbf {A} ={\begin{pmatrix}{\mathit {i}}a&-b+{\mathit {i}}c&-e+{\mathit {i}}f\\b+{\mathit {i}}c&{\mathit {i}}d&-h+{\mathit {i}}k\\e+{\mathit {i}}f&h+{\mathit {i}}k&{\mathit {i}}g\end{pmatrix}}$ is a skew-Hermitian matrix.
Clearly, entries on the main diagonal must be purely imaginary.

References

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