Hermitian matrix

A Hermitian matrix (or self-adjoint matrix) is one which is equal to its Hermitian adjoint (also known as its conjugate transpose). Every entry in the transposed matrix is equal to the complex conjugate of the corresponding entry in the original matrix:

$a_{i,j}^{\mathrm {T} }=a_{j,i}={\overline {a_{i,j}}}$,

or in matrix notation:

$\mathbf {A} ^{\mathrm {T} }={\overline {\mathbf {A} }}\quad \Longrightarrow \quad \mathbf {A} ={\overline {\mathbf {A} ^{\mathrm {T} }}}\equiv \mathbf {A} ^{*}$,

where A^T stands for A transposed. In physics the dagger symbol is often used instead of the star:

$\mathbf {A} ^{\dagger }\equiv {\overline {\mathbf {A} ^{\mathrm {T} }}}$,

so that a physics text would define a Hermitian matrix as a matrix satisfying

$\mathbf {A} =\mathbf {A} ^{\dagger }$.

Forming the Hermitian adjoint

As an example a general 3×3 Hermitian matrix is introduced:

$\mathbf {A} ={\begin{pmatrix}a&b+{\mathit {i}}c&e+{\mathit {i}}f\\b-{\mathit {i}}c&d&h+{\mathit {i}}k\\e-{\mathit {i}}f&h-{\mathit {i}}k&g\end{pmatrix}}\quad {\hbox{with}}\quad a,b,c,d,e,f,g,h,k\in \mathbb {R} .$.

First we form the transpose matrix

$\mathbf {A} ^{\mathrm {T} }={\begin{pmatrix}a&b-{\mathit {i}}c&e-{\mathit {i}}f\\b+{\mathit {i}}c&d&h-{\mathit {i}}k\\e+{\mathit {i}}f&h+{\mathit {i}}k&g\end{pmatrix}}$,

by replacing $A_{i,j}$ with $A_{j,i}$.

Second, we take the complex conjugate of each entry to form the Hermitian adjoint:

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

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. Indeed, by definition $a_{i,i}={\overline {a_{i,i}}}$ which implies $a_{i,i}\in \mathbb {R} .$

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.

Decomposition

Any square matrix C can be written as the sum of a Hermitian matrix A and skew-Hermitian matrix (see below) B:

$\mathbf {C} =\mathbf {A} +\mathbf {B}$

where

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

It follows immediately from the linearity of the Hermitian adjoint that A is Hermitian and B skew-Hermitian:

$(\mathbf {C} \pm \mathbf {C} ^{*})^{*}=\mathbf {C} ^{*}\pm \mathbf {C} =\pm (\mathbf {C} \pm \mathbf {C} ^{*})$

Normal

All Hermitian matrices are normal, i.e. $\mathbf {AA^{*}=A^{*}A}$, and thus the finite dimensional spectral theorem 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

See Also

• Hermitian Operator