All Questions
11 questions
3
votes
1
answer
243
views
Existence of a matrix with bounded entries and large smallest singular value
Is the following statement true?
For all $n$ large enough, there exists an $M_n \in [-1,1]^{n \times n}$ such that the smallest singular value of $M_n$, $\sigma_n(M_n) \gtrsim \sqrt{n}$.
If $n$ is ...
- 261
1
vote
1
answer
684
views
Probability that random Bernoulli matrix is full rank
This is probably known already, but I could not find a quick argument.
Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the ...
- 1,096
7
votes
1
answer
880
views
Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding ...
- 272
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
0
votes
1
answer
162
views
Number of symmetric matrix with fixed margins
I'm looking for the cardinality of the set of symmetric matrices with entries in $\mathbb{Z}^*$ (the nonnegative integers) and fixed margins $\mathbf{k}=(k_1,k_2,...,k_q)$. I've also seen them called ...
- 175
4
votes
0
answers
355
views
Distribution of min/max row sum of matrix with i.i.d. uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $1$.
all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
- 272
21
votes
0
answers
2k
views
The Fourier Transform of taking Eigenvalues
The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
- 24.7k
1
vote
1
answer
394
views
On rank of random $0/1$ matrices
It is known that a $0/1$ matrix picked from uniform distribution from $\{0,1\}^{n\times n}$ is non-singular with probability $1-o(1)$.
Fix an integer $t$.
Consider a random matrix formed the ...
user94040
1
vote
2
answers
477
views
Worst case difference in rank by column-row swapping
Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...
- 13.9k
2
votes
0
answers
59
views
Min/max row-sum distribution of a symmetric matrix of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then ...
- 272
11
votes
1
answer
636
views
A simple proof for a theorem of Szekeres and Turán
Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
- 24.7k