Eigenvalues of real symmetric and hermitian matrices are
Answers
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:
{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}
or in matrix form:
{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}.
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix {\displaystyle A}A is denoted by {\displaystyle A^{\mathsf {H}}}{\displaystyle A^{\mathsf {H}}}, then the Hermitian property can be written concisely as
{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}
Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are {\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}, although note that in quantum mechanics, {\displaystyle A^{\ast }}A^{\ast } typically means the complex conjugate only, and not the conjugate transpose.