# What does the inner product of two complex general vectors have to do with complex conjugation?

$\newcommand{\qr}[1]{|#1 \rangle} \newcommand{\ql}[1]{\langle #1|} \newcommand{\q}[2]{\langle #1 | #2 \rangle} \newcommand{\v}[2]{\langle #1,#2\rangle}$ I'm reading "Quantum computer science, an introduction," N. David Mermin, Cambridge University Press, 2007. In Appendix A, he writes that "in a vector space over the complex numbers the inner product of two general vectors is a complex number satisfying $\q{\psi}{\phi} = \q{\psi}{\phi}^*$, where $*$ denotes complex conjugation." (Page 160.)

I must not have understood this. I tried the following example. What's the inner product of $\v{2+3i}{7+3i} \cdot \v{1+2i}{2+2i}$? It's $4 + 27i$. If I commute $\v{1+2i}{2+2i} \cdot \v{2+3i}{7+3i}$, I get the same $4 + 27i$ because inner products are commutative. Taking the complex conjugate $4 + 27i$, I get $4 - 27i$ which is not the same as its complex conjugate, $4 + 27i$.

So I'm pretty lost here. What does he mean by what he said?

The inner product on a complex vetorspace has conjugate symmetry (so not commutative).

An inner product is a function that assigns to every ordered pair of vectors $$x$$ and $$y$$ in V a scalar in $$\mathbb{C}$$, denoted $$\langle x,y\rangle$$, such that $$\forall\ x, y, z \in V$$ and $$c \in \mathbb{C}$$: $$\begin{equation*} V \rightarrow F: \quad x,y \rightarrow \langle x,y\rangle \end{equation*}$$

$$\begin{array} ((a)\quad \langle x+z,y\rangle= \langle x,y\rangle+ \langle z,y\rangle \\[5pt] (b)\quad \langle cx,y\rangle= c \langle x,y\rangle \\[5pt] (c)\quad \overline{\langle x,y\rangle} =\langle y,x\rangle \\[5pt] (d)\quad \langle x,x\rangle >0 \ \text{if} \ x\ne 0 \\[5pt] \end{array}$$

Here $$\overline{x}$$ denotes the complex conjugate of $$x$$.

In complex n dimensional euclidean space the inner product of two vectors $$(a_1,\ldots,a_n)$$ and $$(b_1,\ldots,b_n)$$ defined by $$\sum_{j=1}^n \overline{a_j}b_j$$

So in your example the correct way of calculation is

$$$$\langle{2+3i},{7+3i}\rangle \cdot \langle{1+2i},{2+2i}\rangle=28+9i$$$$

and

$$$$\langle{1+2i},{2+2i} \rangle \cdot \langle{2+3i},{7+3i}\rangle =28-9i$$$$

which is consistent with $$(c)\ \overline{\langle x,y\rangle} =\langle y,x\rangle$$.

To a physicist, the inner product of vectors $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ is $\sum_{j=1}^n \overline{a_j}b_j$. Also a physicist would write $z^*$ for the complex conjugate of $z$ rather than $\overline z$.