Hermitian function

Short description: Type of complex function

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:

[math]\displaystyle{ f^*(x) = f(-x) }[/math]

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

[math]\displaystyle{ f^*(x_1, x_2) = f(-x_1, -x_2) }[/math]

for all pairs [math]\displaystyle{ (x_1, x_2) }[/math] in the domain of [math]\displaystyle{ f }[/math].

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

• the real part of [math]\displaystyle{ f }[/math] is an even function,
• the imaginary part of [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ f }[/math] is real-valued if and only if the Fourier transform of [math]\displaystyle{ f }[/math] is Hermitian.
• The function [math]\displaystyle{ f }[/math] is Hermitian if and only if the Fourier transform of [math]\displaystyle{ 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 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 [math]\displaystyle{ f \star g = f*g }[/math].

Where the [math]\displaystyle{ \star }[/math] is cross-correlation, and [math]\displaystyle{ * }[/math] is convolution.

• If both f and g are Hermitian, then [math]\displaystyle{ f \star g = g \star f }[/math].

See also

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