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If I have two vectors from $\mathbb{R}^q$ then their inner product gives the length of the projection of one on them onto the other multiplied by the other's length. I have searched but couldn't find an intuitive analoge interpretation for the complex inner product. And why do we need complex inner products?

(I'm learning from the book Linear Algebra Done Right of Axler, and they don't really give a motivation for it either, only for the real case..)

CcVHKakalLLOOPPOkKkkKk
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    Why do you need inner products at all? – amd May 26 '19 at 16:11
  • @amd I'm not sure.. I'm mostly used to work on real finite-dimention vectorial spaces, which all are isomorphic to $R^q$. And in $R^q$ with the euclidean inner-product, it has a nice geometric intrepertation so I can make back and forth between geometry and linear algebra, the former of which I only have very superficial knowledge. – CcVHKakalLLOOPPOkKkkKk May 26 '19 at 16:43
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    Flip this conceptual relationship around—defining an inner product can be viewed as a way of imposing a geometry on the vector space. In particular, some notion of orthogonality is useful regardless of what kinds of objects the vectors are. – amd May 26 '19 at 16:47
  • @amd hmm yes that way to see things is much more satisfying. But while it helps with the real case, the complex inner product still seems arbitrary.. why is it useful to have it $<a,b>$ = $Conjugate( <b,a> )$, and have it spit out a complex number? And how does this geometry with complex scalars work? – CcVHKakalLLOOPPOkKkkKk May 26 '19 at 16:54
  • It makes $\langle a,a\rangle = \lVert a\rVert$ a real number. – amd May 26 '19 at 19:18
  • @amd it does. But there are infinitely other ways to make it a real number. Why is that way better than the others? – CcVHKakalLLOOPPOkKkkKk May 26 '19 at 19:24

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It's needed in quantum mechanics, for measures of overlap between physical states. For example,

  • $P(a\to b)=(a,b)(b,a)/(a,a)(b,b)$ is the Born rule for the probability of transition between two states $a\to b$, and
  • $\mathrm{e}^{2\mathrm{i}\phi(a\to b\to c \to a)}=\frac{(a,c)(c,b)(b,a)}{(a,b)(b,c)(c,a)}$ is a geometric phase acquired during a transition $a\to b\to c \to a$.
K B Dave
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