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In mathematics, an element x of a *-algebra is normal if it satisfies x x ∗ = x ∗ x . {\displaystyle xx^{*}=x^{*}x.}
This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator A from a Hilbert space into itself is called unitary if A A ∗ = A ∗ A , {\displaystyle AA^{*}=A^{*}A,} where the adjoint of A is A and the domain of A is the same as that of A. See normal operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is normal if and only if the matrix describing A with respect to this basis is a normal matrix.
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