A square matrix is called Hermitian if it is self-adjoint. Therefore, a Hermitian matrix is defined as one for which
(1)
|
where
denotes the conjugate transpose. This is equivalent
to the condition
(2)
|
where
denotes the complex conjugate. As a result of
this definition, the diagonal elements
of a Hermitian matrix are real numbers (since
), while other elements may be complex.
Examples of
Hermitian matrices include
(3)
|
and the Pauli matrices
(4)
| |||
(5)
| |||
(6)
|
Examples of
Hermitian matrices include
(7)
|
An integer or real matrix is Hermitian iff it is symmetric.
A matrix
can be tested to see if it is Hermitian in the Wolfram
Language using HermitianMatrixQ[m].
Hermitian matrices have real eigenvalues whose eigenvectors form a unitary basis. For real matrices, Hermitian is the same as symmetric.
Any matrix which is not Hermitian can be expressed as the sum of a Hermitian
matrix and a antihermitian matrix using
(8)
|
Let
be a unitary matrix and
be a Hermitian matrix. Then the adjoint
of a similarity transformation is
(9)
| |||
(10)
| |||
(11)
| |||
(12)
| |||
(13)
|
The specific matrix
(14)
| |||
(15)
|
where
are Pauli matrices, is sometimes called "the"
Hermitian matrix.