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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 Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. [4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs. [5]
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.
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.
Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform: [citation needed] The function is real-valued if and only if the Fourier transform of is Hermitian.
The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N −1; Hermitian operators (i.e., self-adjoint operators): N* = N; skew-Hermitian operators: N* = −N; positive operators: N = MM* for some M (so N is self-adjoint). A normal matrix is the matrix expression of a normal operator on the ...
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
If is a normal element of a C*-algebra , then for every real-valued function, which is continuous on the spectrum of , the continuous functional calculus defines a self-adjoint element (). [ 5 ] Criteria