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YuiTo Cheng
YuiTo Cheng
6:03 AM
@MartinSleziak Can you suggest a more informative title regarding How to prove a certain determinant is nonzero? I try to specify the matrix by its row vector but it still exceeds the character limit.
 
Martin Sleziak
Martin Sleziak
6:38 AM
10
Q: How to prove a certain determinant is non-zero

Ahmadreza MomeniConsider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero? $$ \begin{pmatrix} \frac{1}{\beta + 1} & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{p+1}\\ \frac{1}{\beta + 2} & \frac{1}{3} &...

linear-algebra determinants
This may be a generalized Hilbert matrix. Maybe math.univ-lille1.fr/~otrt/otrt/Aleman.pdf will help. If not, anyway, I've given you a search term. — Gerry Myerson yesterday
I suppose that the title would be too big even with smallmatrix: $\det\left(\begin{smallmatrix}
\frac{1}{\beta + 1} & \frac{1}{2} & \frac{1}{3} & \dots & \frac{1}{p+1}\\
\frac{1}{\beta + 2} & \frac{1}{3} & \frac{1}{4} & \dots & \frac{1}{p+2}\\
\frac{1}{\beta + 3} & \frac{1}{4} & \frac{1}{5} & \dots & \frac{1}{p+3}\\
\vdots & \vdots & \vdots & \dots & \vdots \\
\frac{1}{\beta + p + 1} & \frac{1}{p+2} & \frac{1}{p+3} & \dots & \frac{1}{2p+1}
\end{smallmatrix}\right)$
Hilbert matrix was mentioned in the comments. Maybe "generalized Hilbert matrix" or "Hilbert matrix with a parameter" or something like that?
If you wanted to add MathJax, that will push the question out of the HNQ list - I am not sure whether MO community sees that as a positive or as a negative thing.
titles Does somebody have more suggestions how to improve the title here: How to prove a certain determinant is non-zero.
@YuiToCheng I wrote above what I was able to think of, but I do not have some really good suggestion.
I will add a comment on matrices in the titles from elsewhere:
in CRUDE, Mar 18 '16 at 13:20, by Daniel Fischer
@MartinSleziak In titles (also elsewhere), \begin{smallmatrix} … \end{smallmatrix} is quite useful. With that, $3\times 3$ matrices in titles are acceptable. (But if you can make an informative title without a matrix, that's probably better, since even a smallmatrix takes much vertical space at $3\times 3$.)
 
YuiTo Cheng
YuiTo Cheng
7:26 AM
@MartinSleziak That will do the trick!
@MartinSleziak My observation is MO has an ample supply of HNQs though, at least very recently.
 
Martin Sleziak
Martin Sleziak
7:50 AM
There were several discussions related to HNQs, I have linked to some of them here: meta.mathoverflow.net/questions/linked/4274
My impression from those discussions was that non-negligible part of users do not like MO questions in the HNQ list. (Although the primary concern is probably drive-by voting.)
OTOH, in many cases there are suitable titles which include MathJax - which would prevent the question from entering HNQ. As far as I can tell, I do not see such edits on MO very often.
@YuiToCheng The pending edit on that question was approved, so now it should be possible to edit the question (including possible improvements to the title).
 
YuiTo Cheng
YuiTo Cheng
8:45 AM
@MartinSleziak I submitted the edit. I am not completely sure the matrix is a generalized Hilbert matrix, but it is definite related. That's why I rephrase it as a "Hilbert-like matrix with parameter"
 

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