---
tag: 48P4A1CX85
kind: concept
name: Hermitian form
msc: [15A63, "11E39"]
specializes: [054YEQ0Y8M]
sources:
  - url: https://en.wikipedia.org/wiki/Sesquilinear_form
---

## Summary

A Hermitian form on a complex vector space is a [[sesquilinear form|054YEQ0Y8M]] $h$ with $h(v, u) = \overline{h(u, v)}$ for all vectors $u, v$.
It is the complex analogue of a [[symmetric bilinear form|0AXZNE5XNH]].
The value $h(v, v)$ is always real, so Hermitian forms can be positive, negative or indefinite.
Complex [[inner products|0NB0AWGJBW]] are exactly the positive definite Hermitian forms.

## Explanations

### L1 human

Over the real numbers, symmetric bilinear forms such as the dot product measure squared lengths.
Over the complex numbers, one has to conjugate one argument to keep $h(v, v)$ real; this gives a Hermitian form.
On $\C^n$, the Hermitian forms are exactly $h(x, y) = y^* M x$ with $M$ a [[Hermitian matrix|0P17NGQQMQ]].
For example, $h(x, y) = x_1 \overline{y_1} - x_2 \overline{y_2}$ on $\C^2$ is Hermitian, with $h(v, v) = \abs{v_1}^2 - \abs{v_2}^2$.