↑ By combining the linear in the first argument property with the conjugate symmetry property you get conjugate-linear in the second argument: ⟨ x , b y ⟩ = ⟨ x , y ⟩ b ¯ . {\textstyle \langle x,by\rangle =\langle x,y\rangle {\overline {b}}.} This is how the inner product was originally defined and is still used in some old-school math communities. However, all of engineering and computer science, and most of physics and modern mathematics now define the inner product to be linear in the second argument and conjugate-linear in the first argument because this is more compatible with several other conventions in mathematics. Notably, for any inner product, there is some hermitian, positive-definite matrix M {\textstyle M} such that ⟨ x , y ⟩ = x ∗ M y . {\textstyle \langle x,y\rangle =x^{*}My.} (Here, x ∗ {\textstyle x^{*}} is the conjugate transpose of x . {\textstyle x.} )
↑ This means that ⟨ x , y + z ⟩ = ⟨ x , y ⟩ + ⟨ x , z ⟩ {\displaystyle \langle x,y+z\rangle =\langle x,y\rangle +\langle x,z\rangle } and ⟨ x , a y ⟩ = a ¯ ⟨ x , y ⟩ {\displaystyle \langle x,ay\rangle ={\overline {a}}\langle x,y\rangle } for all vectors x, y, and z and all scalars a.
↑ A bar over an expression denotes complex conjugation; e.g., x ¯ {\textstyle {\overline {x}}} is the complex conjugation of x . {\textstyle x.} For real values, x = x ¯ {\textstyle x={\overline {x}}} and conjugate symmetry is just symmetry.
↑ Recall that for any complex number c, c is a real number if and only if c = c. Using y = x in condition (2) gives ⟨ x , x ⟩ = ⟨ x , x ⟩ ¯ , {\displaystyle \langle x,x\rangle ={\overline {\langle x,x\rangle }},} which implies that ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is a real number.
↑ This is because condition (1) and positive-definiteness implies that ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is always a real number. And as mentioned before, a sesquilinear form is Hermitian if and only if ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is real for all x.
