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:

👁 {\displaystyle f^{*}(x)=f(-x)}

(where the 👁 {\displaystyle ^{*}}
indicates the complex conjugate) for all 👁 {\displaystyle x}
in the domain of 👁 {\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 👁 {\displaystyle f}
is a function of two variables it is Hermitian if

👁 {\displaystyle f^{*}(x_{1},x_{2})=f(-x_{1},-x_{2})}

for all pairs 👁 {\displaystyle (x_{1},x_{2})}
in the domain of 👁 {\displaystyle f}
.

From this definition it follows immediately that: 👁 {\displaystyle f}
is a Hermitian function if and only if

• the real part of 👁 {\displaystyle f}
  is an even function,
• the imaginary part of 👁 {\displaystyle f}
  is an odd function.

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 👁 {\displaystyle f}
  is real-valued if and only if the Fourier transform of 👁 {\displaystyle f}
  is Hermitian.
• The function 👁 {\displaystyle f}
  is Hermitian if and only if the Fourier transform of 👁 {\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 👁 {\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 👁 {\displaystyle f\star g=g\star f}
  .

• Complex conjugate – Fundamental operation on complex numbers
• Even and odd functions – Functions such that f(–x) equals f(x) or –f(x)