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The adjoint may also be called the Hermitian conjugate or simply the Hermitian [1] after Charles Hermite. It is often denoted by A † in fields like physics , especially when used in conjunction with bra–ket notation in quantum mechanics .
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, (), which is also sometimes called adjoint. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } with real entries reduces to the transpose of A {\displaystyle \mathbf {A} } , as the conjugate of a real number is the number itself.
In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: f ∗ ( x ) = f ( − x ) {\displaystyle f^{*}(x)=f(-x)}
Then F*P(D)F is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function P. More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support.
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian adjoint N*, that is: NN* = N*N. [1] Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood.
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
A unitary operator is a bounded linear operator U : H → H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H → H is the identity operator. The weaker condition U*U = I defines an isometry. The other weaker condition, UU* = I, defines a coisometry.
Moreover, the following completeness identity for the above Hermite functions holds in the sense of distributions: = () = (), where δ is the Dirac delta function, ψ n the Hermite functions, and δ(x − y) represents the Lebesgue measure on the line y = x in R 2, normalized so that its projection on the horizontal axis is the usual Lebesgue ...