A square matrix is called Hermitian if it is self-adjoint. Therefore, a Hermitian matrix is defined as one for which
(1)
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where denotes the conjugate transpose. This is equivalent to the condition
(2)
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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)
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and the Pauli matrices
(4)
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(5)
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(6)
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Examples of Hermitian matrices include
(7)
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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)
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Let be a unitary matrix and be a Hermitian matrix. Then the adjoint of a similarity transformation is
(9)
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(10)
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(11)
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(12)
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(13)
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The specific matrix
(14)
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(15)
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where are Pauli matrices, is sometimes called "the" Hermitian matrix.