Also known as Hermitian matrix and self-adjoint matrix.
It is a complex matrix such that coincides with its conjugate transpose. They play an analogous role to symmetric matrixs. That is, all of them are matrices which can be obtained by a real diagonal matrix after a base change by means of a unitary change of basis (see spectral theory).
In other words, if you find an Hermitian matrix what you are dealing with is with a (real) scale transformation in orthogonal directions but expressed in a basis obtained applying a unitary transformation to the canonical basis. That is the reason why self adjoint matrices have real eigenvalues and orthogonal eigenvectors.
They are generalized to self adjoint operators.
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Author of the notes: Antonio J. Pan-Collantes
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