How does the hermitian sesquilinear form look like?
I mean some concrete example. I only found the abstract definitions but I would like to try diagonalize such forms or look at its signature, see the difference compared to the classic billinear forms.
In a two-dimensional hermitian space: $$L(u,v)=a_{11}\bar{u}_1 v_1 + a_{12}\bar{u}_1 v_2 + a_{21}\bar{u}_2 v_1 + a_{22}\bar{u}_2 v_2.$$ Here the matrix $A=(a_{ij})$ is hermitian: $$a_{ji} = \bar{a}_{ij}.$$ (Please observe that this property implies that the diagonal elements of $A$ are real numbers). In matrix notation: $L(u,v)=u^H A v$, where $u^H$ is the conjugate transpose of the column vector $u$. The quadratic form associated to $L$ is $$Q(u)=L(u,u)=u^H A u = a_{11}|u_1|^2 + a_{12}\bar{u}_1 u_2 + a_{21}\bar{u}_2 u_1 +a_{22}|u_2|^2 = a_{11}|u_1|^2 + 2 Re(a_{12}\bar{u}_1 u_2) + a_{22}|u_2|^2 .$$ In the last line we have exploited the relation $a_{21} = \bar{a}_{12}$: $$ a_{12} \bar{u}_1 u_2 + a_{21} \bar{u}_2 u_1 = a_{12} \bar{u}_1 u_2 + \bar{a}_{12} \cdot \overline{\bar{u}_1 u_2} = a_{12} \bar{u}_1 u_2 + \overline{a_{12} \bar{u}_1 u_2} = 2 Re(a_{12}\bar{u}_1 u_2).$$
Now it is up to you to generalize this object to $n$-dimensional hermitian spaces and also to apply the spectral theorem to the hermitian matrix $A$ in such a way to get to the canonical form of $Q(u)$.