In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose. That is, if the element in the j {\displaystyle j} -th row and k {\displaystyle k} -th column of a Hermitian matrix A {\displaystyle A} is some complex number A j k = x + i y {\displaystyle A_{jk}=x+iy} , then the element in the k {\displaystyle k} -th row and j {\displaystyle j} -th column is its complex conjugate A k j = A j k ¯ = x − i y {\displaystyle A_{kj}={\overline {A_{jk}}}=x-iy} , for every pair of indices j {\displaystyle j} and k {\displaystyle k} . Hermitian matrices can be understood as the complex generalization of symmetric real matrices. Using the notation A T {\displaystyle \textstyle A^{\mathsf {T}}} to mean the transpose of A {\displaystyle A} and an overline to mean the entrywise complex conjugate of a matrix, the Hermitian property is equivalent to the equality
A = A T ¯ . {\displaystyle A={\overline {A^{\mathsf {T}}}}.}
The conjugate transpose of a matrix is often denoted A H {\displaystyle \textstyle A^{\mathsf {H}}} , in terms of which the Hermitian property can be more concisely expressed as A = A H {\displaystyle \textstyle A=A^{\mathsf {H}}} . Equivalent notations in common use include A H {\displaystyle \textstyle A^{\mathsf {H}}} , A † {\displaystyle \textstyle A^{\dagger }} , and A ∗ {\displaystyle \textstyle A^{*}} , although in quantum mechanics, A ∗ {\displaystyle \textstyle A^{*}} typically means the complex conjugate only, and not the conjugate transpose. Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share with symmetric real matrices the property of always having real eigenvalues.
Alternative characterizations Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
Equality with the adjoint A square matrix A {\displaystyle A} is Hermitian if and only if it is equal to its conjugate transpose, that is, for any pair of vectors v , w {\displaystyle \mathbf {v} ,\mathbf {w} } , it satisfies
⟨ v , A w ⟩ = ⟨ A v , w ⟩ , {\displaystyle \langle \mathbf {v} ,A\mathbf {w} \rangle =\langle A\mathbf {v} ,\mathbf {w} \rangle ,}
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denotes the standard inner product operation in complex coordinate space, a Hermitian form defined by ⟨ v , w ⟩ = v H w {\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle =\mathbf {v} ^{\mathsf {H}}\mathbf {w} } . This is also the way that the more general concept of self-adjoint operator is defined.
Real-valuedness of quadratic forms An n × n {\displaystyle n\times n} matrix A {\displaystyle A} is Hermitian if and only if
⟨ v , A v ⟩ ∈ R , for all v ∈ C n . {\displaystyle \langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad {\text{for all }}\mathbf {v} \in \mathbb {C} ^{n}.}
Spectral properties A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.
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