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)}
(where the ∗ {\displaystyle ^{*}} indicates the complex conjugate) for all x {\displaystyle x} in the domain of f {\displaystyle f} . In physics, this property is referred to as PT symmetry. This definition extends also to functions of two or more variables, e.g., in the case that f {\displaystyle f} is a function of two variables it is Hermitian if
f ∗ ( x 1 , x 2 ) = f ( − x 1 , − x 2 ) {\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})}
for all pairs ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} in the domain of f {\displaystyle f} . From this definition it follows immediately that: f {\displaystyle f} is a Hermitian function if and only if
the real part of f {\displaystyle f} is an even function, the imaginary part of f {\displaystyle f} is an odd function.
Motivation Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:
The function f {\displaystyle f} is real-valued if and only if the Fourier transform of f {\displaystyle f} is Hermitian. The function f {\displaystyle f} is Hermitian if and only if the Fourier transform of f {\displaystyle f} is real-valued. Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal. Informally, only half of the fourier transform of a real signal is needed to lossessly represent it in frequency domain. For the magnitude spectra (obtained from DFT), the axis of symmetry is around the Nyquist point; one half is the mirror image of the other.
If f is Hermitian, then f ⋆ g = f ∗ g {\displaystyle f\star g=f*g} . Where the ⋆ {\displaystyle \star } is cross-correlation, and ∗ {\displaystyle *} is convolution.
If both f and g are Hermitian, then f ⋆ g = g ⋆ f {\displaystyle f\star g=g\star f} .
See also Complex conjugate – Fundamental operation on complex numbers Even and odd functions – Functions such that f(–x) equals f(x) or –f(x)