In the context of linear algebra and functional analysis, a self-adjoint operator (also known as a self-adjoint matrix in finite dimensions) is a specific type of linear operator that has a particular property regarding its adjoint.
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A densely defined operator on a Hilbert space is self-adjoint when it equals its adjoint operator, including equality of their operator domains:For a bounded linear operator defined on the whole Hilbert space, this is equivalent tofor all vectors . For an unbounded self-adjoint operator, the domain condition is essential: the inner-product identity on alone gives only a symmetric operator.
