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I am having some trouble with the inner product and the Gram-Schmidt process for complex vectors as I am trying to learn it on my own. This is mainly due to the discrepancy with my text book and what I found online.

1) inner product:

if you refer to: What is the dot product of complex vectors?

you would notice that the author of the answer wrote that $$\mathbf{A} \cdot \mathbf{B} = \sum_i a_i \bar{b_i} $$

Where I understand that this implies that: $$\mathbf{A} \cdot \mathbf{B} = \mathbf{A} \cdot \mathbf{B}^* $$

Where $\mathbf{B}^*$ is the complex conjugate of the matrix B

My textbook however, wrote this:

$$\langle \mathbf{A} \mid \mathbf{B} \rangle = \sum_{i=1}^N a_i^*b_i $$

which kind of contradicts what i found online. So who is correct?

2) The Gram-Schmidt for complex vectors

My textbook writes that if the vectors $\mathbf{x_i}$ are not mutually orthogonal, and we want to construct new vectors $\mathbf{z_i}$ that are orthogonal, the process is:

$$z_1 = x_1$$ $$z_2 = x_2 - [(\hat{z_1}^\dagger)x_2]\hat{z_1} $$ $$...$$

for $z_2$, I understand that:

$$[(\hat{z_1}^\dagger)x_2]\hat{z_1} = \frac{{z_1^\dagger}{x_2}}{\|z_1\|^2} z_1 = \frac{\langle \mathbf{z_1}\mid\mathbf{x_2}\rangle}{\langle \mathbf{z_1}\mid\mathbf{z_1} \rangle} \mathbf{z_1}$$

But if we refer to wikipedia: https://en.wikipedia.org/wiki/Gram–Schmidt_process

We can see that wiki clearly defines

$$\mathbf{u_2} = \mathbf{v_2} - \operatorname{proj}_\mathbf{u_1} (\mathbf{v_2})=\mathbf{v_2}-\frac{\langle \mathbf{v_2} \mid \mathbf{u_1} \rangle}{\langle \mathbf{u_1} \mid \mathbf{u_1} \rangle} \mathbf{u_1}$$

which in my textbook example, would be:

$$\mathbf{z_2} = \mathbf{x_2} - \operatorname{proj}_\mathbf{z_1} (\mathbf{x_2})=\mathbf{x_2}-\frac{\langle \mathbf{x_2} \mid \mathbf{z_1} \rangle}{\langle \mathbf{z_1} \mid \mathbf{z_1} \rangle} \mathbf{z_1}$$

So you can see, if both the wiki and my textbook is correct, then this implies that:

$${\langle \mathbf{z_1}\mid\mathbf{x_2}\rangle} = {\langle \mathbf{x_2}\mid \mathbf{z_1} \rangle} $$

But this does not make sense! Because:

$${\langle \mathbf{z_1}\mid\mathbf{x_2}\rangle} = z_1^\dagger x_2 \neq x_2^\dagger z_1 = {\langle \mathbf{x_2} \mid \mathbf{z_1} \rangle} $$

For both questions, I don't think anyone is wrong, rather, I feel that I may be missing something.

Sorry if both are stupid questions, but thanks in advance!

Bernard
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D. Soul
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  • The scalar product satisfies $\langle x,ry\rangle=\overline{r}\langle x,y\rangle$ and $\langle x,y\rangle=\overline{\langle y, x\rangle}$. When the scalars are real these reduce to $\langle x,ry\rangle=r\langle x,y\rangle$ and $\langle x,y\rangle=\langle y, x\rangle$, respectively. Whichever formula that you found that is missing the conjugates it must be because they are assuming real scalars. – logarithm May 24 '19 at 14:00
  • I think strictly speaking these are both satisfactory inner products, however I have never seen anybody define $\langle A \mid B \rangle$ with the conjugate on the first component as your textbook does, before. – Jack Crawford May 24 '19 at 14:04
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    @logarithm Thanks for your reply! but in my textbook example as you see above, there is a dagger symbol, which I understand is the hermitian or the transpose of the conjugate. Doesn't this mean that the textbook is NOT assuming real scalars? – D. Soul May 24 '19 at 14:18
  • @JackCrawford Does this mean both are correct? But the output would not be equal though, since $$\sum_{i}a_ib_i^* \neq \sum_{i}a_i^*b_i$$ – D. Soul May 24 '19 at 14:19
  • @tzyyyy As it sometimes tends to be with a lot of these sorts of definitions in math, there may be multiple ways of constructing something that fits the definitions which functions equivalently. Of course, wherever you see the inner product $\langle a, b\rangle$ in your textbook, you translate that to any other textbook as $\langle a^, b^ \rangle$, and otherwise everything else still works the same. I know at the very least that there are multiple choices for how you construct the inner product in whether you choose for it to be sesquilinear or not, perhaps this is another arbitrary choice. – Jack Crawford May 24 '19 at 14:23
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    Some people assume that the semi-linear component is the first one instead of the second one. The text with the complex conjugate in the first component must be assuming that. – logarithm May 24 '19 at 14:24
  • @tzyyyy The most I can say is that while the choice your textbook seems to use may very well be internally consistent, I have at least never seen it used anywhere else, and I think the definition using $$\sum_{i} a_i b^*_i$$ is more conventional. It's quite possibly still a valid choice of definition, though (as @ logarithm notes, and I'll take their word for it), but you will need to be careful when translating it elsewhere. – Jack Crawford May 24 '19 at 14:24
  • @JackCrawford Hi guys thanks for your help. Apparently I just got word from my friends that the reason for confusion is because my textbook is a physics textbook, so it defines the complex conjugate to be the first term of the inner product instead of the 2nd term. This is the reason why the gran schmidt is flipped compared to wikipedia.

    But thank you nonetheless!

     – D. Soul May 24 '19 at 14:40

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