3

I don't know why I'm confused this elementary stuff, but is the following correct?

$v_{1}v_{1}^{*} + v_{1}v_{2}^{*}... + v_{1}v_{n}^{*} + v_{2}v_{1}^{*} + v_{2}v_{2}^{*} ... + v_{2}v_{n}^{*} + v_{n}v_{1}^{*} + v_{n}v_{2}^{*} + ... + v_{n}v_{n}^{*} = \Sigma_{i=1}^{n}\Sigma_{j=1}^{n}v_{i}v_{j}^{*}$

where $v^{*}$ is the conjugate of the vector $v$?

Lory
  • 203
  • 1
  • 6
  • that's correct! – D S Apr 15 '23 at 16:38
  • 1
    Yes, your expression is correct. In general, for any two vectors $v$ and $w$ in a complex vector space, the product $vw^$ represents the entrywise product of $v$ and the conjugate of $w$. The expression you provided is the sum of all possible pairwise entrywise products between the vectors $v_i$ and their conjugates $v_j^$ for $i, j = 1, \dots, n$. To write it in a more compact form, you can use the double summation notation: $\sum_{i=1}^n \sum_{j=1}^n v_i v_j^*$ – al-Hwarizmi Apr 15 '23 at 16:41
  • I’m voting to close this question because it was answered in the comments. – Accelerator Apr 18 '23 at 23:40

1 Answers1

2

It is easier to see if we write it in a series of steps.

$\begin{array}\\ s(n) &=v_{1}v_{1}^{*} + v_{1}v_{2}^{*}... + v_{1}v_{n}^{*} + v_{2}v_{1}^{*} + v_{2}v_{2}^{*} ... + v_{2}v_{n}^{*} + v_{n}v_{1}^{*} + v_{n}v_{2}^{*} + ... + v_{n}v_{n}^{*}\\ &=v_{1}v_{1}^{*} + v_{1}v_{2}^{*}... + v_{1}v_{n}^{*}\\ &\quad + v_{2}v_{1}^{*} + v_{2}v_{2}^{*} ... + v_{2}v_{n}^{*}\\ &\quad ...\\ &\quad + v_{n}v_{1}^{*} + v_{n}v_{2}^{*} + ... + v_{n}v_{n}^{*}\\ &=(v_{1}v_{1}^{*} + v_{1}v_{2}^{*}... + v_{1}v_{n}^{*})\\ &\quad + (v_{2}v_{1}^{*} + v_{2}v_{2}^{*} ... + v_{2}v_{n}^{*})\\ &\quad ...\\ &\quad + (v_{n}v_{1}^{*} + v_{n}v_{2}^{*} + ... + v_{n}v_{n}^{*})\\ &=(\sum_{j=1}^n v_{1}v_{j}^{*})\\ &\quad + (\sum_{j=1}^n v_{2}v_{j}^{*})\\ &\quad ...\\ &\quad + (\sum_{j=1}^n v_{n}v_{j}^{*})\\ &= \Sigma_{i=1}^{n}\Sigma_{j=1}^{n}v_{i}v_{j}^{*}\\ \end{array} $

marty cohen
  • 107,799
  • 1
    It's probably even easier to see if you go the opposite way ;) – Gareth Ma Apr 15 '23 at 18:29
  • True, but I wanted to start with the original expression. I generally dislike writing expressions with "..." as you can see if you look at some of my answers. – marty cohen Apr 15 '23 at 18:32