Problem with Einstein summation notation

Question

Well I've got a bit problem with ESN so let me introduce my problem :

I have in my lesson the demonstration of this propriety without ESN :

If $(x_i)_{1 \le i \le n}$ is an orthogonal familly, then $$ \left|\left| \sum_{i=1}^{n} x_i \right|\right|^2 = \sum_{i=1}^{n} ||x_i||^2 $$

Now my problem is to note this two part of equality with ESN :/

$$ ||x^i||^2= ??? $$

And I also want to know how to note

$$ \sum_{i=1}^n\left(x_i \mid \sum_{j=1}^n x_j\right) $$

where $(.|.)$ is dot product :)

Shadock

Answer 1

Update:

Sorry if there was any confusion from my initial post. I've edited this to more fully explain my answer. If a vector $x$ can be written as $x = (x_1, \ldots, x_k)$ then in Einstein summation notation $x$ can be denoted as $x_i$ if it's covariant or $x^i$ if it's contravariant (I'm not assuming the Lorentz metric for this kind of vector). The norm of $x$ can then be written as $||x||^2 = x^ix_i$ where the summation is implied by the double index. To write it out fully, we have

$$ \underbrace{x^ix_i}_{Not \; components} \;\; =\;\; \sum_{j=1}^n \underbrace{x_j^2}_{components} \;\; =\;\; ||x||^2. $$

Unfortunately if you have multiple vectors $\textbf{x}^{(1)}, \ldots, \textbf{x}^{(m)}$ where I use the bold-face for emphasis, then I don't think there is a way to express the quantity

$$ \sum_{j=1}^m ||\textbf{x}^{(j)}||^2 $$

In terms of the Einstein summation convention. As Zhen Lin pointed out in the comments below, the notation convention is used to manipulate the components of a vector, not multiple vectors at one time.

@Zhen - thanks for your critique. I would've missed this otherwise.

Answer 2

If your vectors are in an euclidean space then you could also write your identity as:

\begin{equation} (x_i)_a(x_j)_a = \frac{1}{n} \delta_{ij} (x_k)_a (x_k)_a \end{equation}

Your second expression can't really be put in ENS, there are no repeated indices there. You could consider, as above, that the inner product is eculidean, and hence I think the best you could do for your second expression would be:

\begin{equation} \sum_{i,j =1}^{n} (x_i)_a (x_j)_a \end{equation}