TITLE: Unsure about a maths symbol
QUESTION [7 upvotes]: Help, help, help!
I've come across this maths symbol,
$[n(i,j)]^{0.5}$ where $n$ is a square matrix.
Does this mean that it is the $(i,j)$ element of $n^{0.5}$? or $n(i,j)^{0.5}$?
source: http://outobox.cs.umn.edu/Random_Walks_Collaborative_Recommendation_Fouss.pdf
2nd page, fifth line from the top of left column:



Thanks a million in advance!
Eh... after looking at some notation he used at page 367, my guess is element-wise square-root?

REPLY [3 votes]: I don't see where it is defined explicitly.  It looks like the square root of the $(i,j)$ element of $n$, as opposed to the $(i,j)$ element of the square root of $n$.  I say that because he justifies that the elements are positive and therefore have square roots, but doesn't seem to remark that the matrix will have a square root.  It will, but I would have expected it to be mentioned if needed.

REPLY [2 votes]: Section 5.1. (on page 5) has the following formula:
$$n(i,j) = V_G (x'_i - x'_j)^T (x'_i - x'_j) = V_G \| x'_i - x'_j \|^2,$$
so I'd say that $[n(i,j)]^{1/2}$ stands for an ordinary element-wise square root, i.e.,
$$[n(i,j)]^{1/2} = \sqrt{n(i,j)} = \| x'_i - x'_j \|\sqrt{V_G}.$$
Value $V_G$ is defined in section 4.2 as the volume of the graph $V_G = \sum_{k=1}^n d_{kk}$, i.e., a nonnegative number, so it has a square root.