stylistic:tone, linguistic:paraphrasing
Complex coordinate space

The general form of an inner product on C n {\displaystyle \mathbb {C} ^{n}} is known as the Hermitian form and is given by ⟨ x, y ⟩ = y † M x = x † M y ¯, {\displaystyle \langle x,y\rangle =y^{\dagger }\mathbf {M} x={\overline {x^{\dagger }\mathbf {M} y}},} where M {\displaystyle M} is any Hermitian positive-definite matrix and y † {\displaystyle y^{\dagger }} is the conjugate transpose of y. {\displaystyle y.} For the real case, this corresponds to the dot product of the results of directionally-different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation.