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:

(where the indicates the complex conjugate) for all in the domain of . 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 is a function of two variables it is Hermitian if

for all pairs in the domain of .

From this definition it follows immediately that: is a Hermitian function if and only if

- the real part of is an even function,
- the imaginary part of 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:^{[citation needed]}

- The function is real-valued if and only if the Fourier transform of is Hermitian.
- The function is Hermitian if and only if the Fourier transform of 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.

- If
*f*is Hermitian, then .

Where the is cross-correlation, and is convolution.

- If both
*f*and*g*are Hermitian, then .

## See also

- Complex conjugate
- Even and odd functions – 1 = Mathematical functions such that f(-x) = f(x) (even) or f(-x) = -f(x) (odd)