# Hermitian function

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{{Short description|Type of complex function}}{{Unreferenced|date=October 2025}}

In [mathematical analysis](/source/mathematical_analysis), a '''Hermitian function''' is a [complex function](/source/Complex-valued_function) with the property that its [complex conjugate](/source/complex_conjugate) is equal to the original function with the variable changed in [sign](/source/sign_(mathematics)):

:<math>f^*(x) = f(-x)</math>

(where the <math>^*</math> indicates the complex conjugate) for all <math>x</math> in the domain of <math>f</math>. In [physics](/source/physics), this property is referred to as [PT symmetry](/source/Non-Hermitian_quantum_mechanics). 

This definition extends also to functions of two or more variables, e.g., in the case that <math>f</math> is a function of two variables it is Hermitian if

:<math>f^*(x_1, x_2) = f(-x_1, -x_2)</math>

for all pairs <math>(x_1, x_2)</math> in the domain of <math>f</math>.

From this definition it follows immediately that: <math>f</math> is a Hermitian function [if and only if](/source/if_and_only_if)

* the real part of <math>f</math> is an [even function](/source/even_function), 
* the imaginary part of <math>f</math> is an [odd function](/source/odd_function).

== Motivation ==
Hermitian functions appear frequently in mathematics, physics, and [signal processing](/source/signal_processing).  For example, the following two statements follow from basic properties of the Fourier transform:{{Citation needed|reason=Proof is not entirely trivial.|date=March 2018}}

* The function <math>f</math> is real-valued if and only if the [Fourier transform](/source/Fourier_transform) of <math>f</math> is Hermitian.
* The function <math>f</math> is Hermitian if and only if the [Fourier transform](/source/Fourier_transform) of <math>f</math> 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](/source/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](/source/Discrete_Fourier_transform)), the axis of symmetry is around the [Nyquist point](/source/Nyquist_rate); one half is the mirror image of the other.

* If ''f'' is Hermitian, then <math>f \star g = f*g</math>.

Where the <math> \star </math> is [cross-correlation](/source/cross-correlation), and <math> * </math> is [convolution](/source/convolution).

* If both ''f'' and ''g'' are Hermitian, then <math>f \star g = g \star f</math>.
<!--------
  An example wanted for these two statements above!
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== See also ==

* {{annotated link|Complex conjugate}}
* {{annotated link|Even and odd functions}}

Category:Types of functions
Category:Calculus

{{Complex numbers}}

{{mathanalysis-stub}}

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Adapted from the Wikipedia article [Hermitian function](https://en.wikipedia.org/wiki/Hermitian_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hermitian_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.