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I am a bit confused regarding the correct formula for the complex inner product of two complex vectors. The textbook I learnt from defines the complex inner product of the vectors: $u$ and $v$ as follows: $$\left ( cu,v \right )=\bar{c}\left ( u,v \right )$$ and $$\left ( u,cv \right )=c\left ( u,v \right )$$ where $c$ is a scalar.

However, in a different textbook, I found the following convention: $$\left ( cu,v \right )=c\left ( u,v \right )$$ and $$\left ( u,cv \right )=\bar{c}\left ( u,v \right )$$

Which one of the above formulas is the one I should use?

Also, for the complex dot product: In some textbooks, I found: $$u\cdot v=\bar{u_{1}}v_{1}+\bar{u_{2}}v_{2}+\cdots+\bar{u_{n}}v_{n}.$$ In other textbooks, I found the following: $$u\cdot v= u_{1}\bar{v_{1}}+u_{2}\bar{v_{2}}+\cdots+u_{n}\bar{v_{n}}$$

  • Which one I should as the definition of the complex dot product? Also, which one is equivalent to $u^{*}v$?
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Either one works; it depends on your context. Mathematicians prefer the second one (linear in the first coordinate, conjugate linear in the second). I believe physicists prefer the former (linear in the second coordinate, conjugate linear in the first coordinate). So long as you adapt formulas appropriately, they give you the same results. – Arturo Magidin Feb 24 '12 at 20:35
If you write your vectors as columns, $u=(u_1,\ldots,u_n)^t$ and $v = (v_1,\ldots,v_n)^t$, then $u^* = (\overline{u_1},\ldots,\overline{u_n})$, so $u^*v$ is given by your first definition of "complex dot product." – Arturo Magidin Feb 24 '12 at 20:37
up vote 2 down vote accepted

"True" is not the right word to use here. The two conventions are interchangeable, since given any inner product of the first kind defining $(u, v)_{\text{new}} = (v, u)_{\text{old}}$ gives an inner product of the second kind. So it doesn't matter which one you use, really, as long as you are consistent.

Nevertheless, I would like to argue that the convention where the inner product is conjugate-linear in the first variable is the "better" one to use. One bit of evidence comes from the fact that $\dim \text{Hom}(V, W)$ naturally appears in the representation theory of finite groups, and it gives an inner product on characters which is conjugate-linear in the first variable. The discussion in the beginning of Baez's Higher-Dimensional Algebra II: 2-Hilbert Spaces is relevant.

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Unfortunately, there are two different conventions floating around out there. Either one is correct, but you need to be very careful that you define everything which depends on the inner product in a manner consistent with the particular definition of inner product you are using. For example, if you use your first definition of inner product then $$u\cdot v=\sum_{i=1}^n\langle u_ie_i,v_ie_i\rangle = \sum_{i=1}^n \bar{u_i}\cdot v_i$$ where $e_1,\ldots,e_n$ is the standard basis, while if you use the second then $$u\cdot v=\sum_{i=1}^n\langle u_ie_i,v_ie_i\rangle = \sum_{i=1}^n u_i\cdot \bar{v_i}$$ and similar modifications have to be made to other definitions and theorems. The moral: be sure to check what definition your professor and textbook use (hope that they are the same!) and if you every become a brilliant mathematician and invent some new math, insist on just one convention.

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