Hermitian form

Remarks

Example: on $\C^2$, let $h(x, y) = x_1 \overline{y_1} + 2 x_2 \overline{y_2}$. It is sesquilinear, and $h(y, x) = y_1 \overline{x_1} + 2 y_2 \overline{x_2} = \overline{h(x, y)}$, so it is Hermitian. Non-example: $h(x, y) = x_1 \overline{y_2}$ on $\C^2$ is sesquilinear but not Hermitian, because $h(e_1, e_2) = 1$ while $h(e_2, e_1) = 0$. Taking $u = v$ gives $h(v, v) = \overline{h(v, v)}$, so $h(v, v)$ is real for every $v$. On $\C^n$, every Hermitian form is $h(x, y) = y^* M x$ for one Hermitian matrix $M$, that is, $M^* = M$. Some authors take the form conjugate-linear in the first argument instead; the condition $h(v, u) = \overline{h(u, v)}$ is the same.