# Meaning of superscript (1) in “$v_1(M) = \arg\min_x \|M - xx^T\|_2, x^{(1)}\ge0$”

Quoted from section 4.1 in the paper Aggregating Crowdsourced Binary Ratings:

... We will also denote the (scaled) top eigenvector of a matrix $$M$$ as $$v_1(M) = \arg\min_x \|M - xx^T\|_2, x^{(1)}\ge0$$.

I understand the concept of scaled leading eigenvector, but I don't understand the constraint $$x^{(1)}\ge0$$. Specifically, what is the meaning of the superscript $$(1)$$?

My first guess was that it means the first entry of $$x$$, but in the rest of the paper, they used the standard notation $$x_i$$ to represent vector entries, so I don't think this is the case.

Edit: Rodrigo de Azevedo suggests in the comments that different co-authors of the paper might use different notations, and perhaps it does mean $$x_1\ge0$$. If this is the case, I have the following questions:

• Have you seen any paper (in applied mathematics and related fields) that uses this weird notation?
• What is the reason behind the $$x_1\ge0$$ requirement? Can we somehow relate the $$v_1(M)$$ in this definition with the leading eigenvector in the "standard" definition $$\arg\max_x \frac{x^TMx}{\|x\|}$$?
• Have you tried contacting the authors? – Rodrigo de Azevedo Jan 4 at 9:17
• @RodrigodeAzevedo Not yet, but I've found a few typos and unrigorous proofs in the paper, so I guess it's some kind of mistake :( – hklel Jan 4 at 21:09
• Four authors. Perhaps one started using $x^{(1)}$ to denote the first entry of vector $x$, and then some other decided to use $x_1$ instead. And they forgot to change the notation in one instance. Perhaps. – Rodrigo de Azevedo Jan 4 at 21:23
• @RodrigodeAzevedo Thanks for the reply! If that's the case, can you take a look at the updated question? – hklel Jan 4 at 22:03

My guess is that it is the first component of the vector. This is so that the problem has a more unique solution since $$xx^T = (-x)(-x)^T$$. This can still fail if $$x^{(1)}=0$$ but depending on the matrices involved this could be very unlikely