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Moments from characteristic function for matrices

When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
user3826143's user avatar
2 votes
1 answer
81 views

Distribution of scaled Johnson-Lindenstrauss transforms

Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that $$ \...
Nuno's user avatar
  • 269
2 votes
1 answer
231 views

Trace inverse of random PSD matrix?

Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E_{A}(\mathrm{Tr}( (A^T A + \lambda \...
goku's user avatar
  • 25
2 votes
0 answers
106 views

The distribution of eigenvalues of linear combinations of random unitary matrices

Suppose that $\alpha_{1},\dots,\alpha_{r}$ are non-zero complex numbers. Let $U_{1},\dots,U_{r}$ be random $n\times n$-unitary matrices. Let $A=\alpha_{1}U_{1}+\dots+\alpha_{r}U_{r}$. I have observed ...
Joseph Van Name's user avatar
2 votes
1 answer
236 views

How can I prove a randomly generated matrix has distinct non-zero eigenvalues?

Consider the following $M×M$ matrix $$ \mathbf A=\sum_{k=1}^K =a_k \mathbf h_k \mathbf h_k^H,(M≥K) $$ where $a_k$'s are real values and $h_k$'s are $M×1$ randomly generated vectors, e.g., complex ...
WPCN's user avatar
  • 31
2 votes
0 answers
95 views

Maximum volume submatrices of a Khatri-Rao product of matrix exponentials

My question requires quite a bit of setup, which leads to a conjecture. So I split my question into three parts, Setup, Conjecture, and Question. Setup: Pick any two right stochastic matrices $\...
Jandré Snyman's user avatar
1 vote
0 answers
225 views

Distribution and expectation of inverse of a random Bernoulli matrix

This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
Nishant Singh's user avatar
2 votes
1 answer
265 views

Limit law of eigenvalue of random matrix with mean different to 0

If $X$ denotes a $m \times n$ random matrix whose entries are independent identically distributed random variables with mean $\mu$ and $\sigma^2 < \infty$, let $$Y = X X^T$$ with $X^T$ the ...
Vu Thanh Tung's user avatar
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{...
VS.'s user avatar
  • 1,826
0 votes
0 answers
47 views

"Probability" for a partitioned matrix to be singular

Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix $$ M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
Ludwig's user avatar
  • 2,712
1 vote
1 answer
218 views

Is there a bound on the norm of the product of second moment matrix with random vector?

Let $X_1,\dots,X_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell_2$ ball, but outside the ball of radius $r$ for some $r \in (0,1)$, i.e. $r \leq \|X_i\| \leq 1$ ....
good bandit's user avatar
2 votes
1 answer
280 views

Properties of eigenvalues and eigenvectors of a particular random matrix

Let $\mathbf{A}$ be a given $n \times m$ matrix with positive entries, and $\mathbf{B}_{n\times m}$ be a random i.i.d complex Gaussian matrix with unit variance. Assume that $\mathbf{C}$ is the ...
Math_Y's user avatar
  • 287
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 ...
hookah's user avatar
  • 1,096
7 votes
1 answer
856 views

Trace of inverse of random positive-definite matrix in high dimension?

Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
Goulifet's user avatar
  • 2,306
4 votes
1 answer
346 views

Rank of a random sparse matrix with nonnegative reals

I believe this should be some standard result in random matrices theory, but my initial search failed to find a definitive answer. The question is given a random sparse matrix $M\in\mathbb{R}^{n\...
jaco's user avatar
  • 161
6 votes
1 answer
1k views

Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
JJJZZZZZ's user avatar
  • 380
4 votes
1 answer
372 views

Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
Simon Segert's user avatar
4 votes
1 answer
3k views

Approximating the expectation of a matrix inverse

Let $$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$ where $A$ is a given $n \times m$ matrix (where $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
Christo's user avatar
  • 67
6 votes
2 answers
738 views

Probability of a large random integer Matrix to have zero determinant

Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$ I would like to ...
Hipstpaka's user avatar
  • 355
2 votes
0 answers
64 views

Largest eigenvalue of two types of slightly different random matrices

Consider two types of slightly different $n \times n$ symmetric random matrices $X$. The diagonal elements of $X$ are fixed as $1$. Suppose $\frac{k}{n} \to \alpha$ for some constant $\alpha\in(0,1)$. ...
Tony's user avatar
  • 272
3 votes
2 answers
581 views

Largest eigenvalue of the adjacency matrix of weighted random graph

I find the theorem for largest eigenvalue of the adjacency matrix of ER random graph in here https://arxiv.org/pdf/math/0106066.pdf. The adjacency matrix is a symmetric random matrix s.t. diagonal ...
Tony's user avatar
  • 272
3 votes
0 answers
151 views

Largest eigenvalue divided by $n$

Let $X$ be an $n\times n$ symmetric random matrix whose diagonal is fixed as $1$, and every element in the upper triangle (excluding the diagonal) is drawn from Bernoulli($p$). The elements in the ...
Tony's user avatar
  • 272
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 ...
Tony's user avatar
  • 272
3 votes
0 answers
414 views

Eigenvalue distribution of a special symmetric matrix of uniform random variables

Given a $n\times n$ symmetric random matrix such that all diagonal elements are all fixed as $0$. all other elements in the upper triangle are uniform random variables over $[0,1]$. all ...
Tony's user avatar
  • 272
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 ...
Tony's user avatar
  • 272
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 ...
Tony's user avatar
  • 272
3 votes
2 answers
2k views

Expected value of the largest singular value of a random matrix with entries in $N (0,1)$

Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest ...
wenyuz's user avatar
  • 33
1 vote
1 answer
552 views

Probability of positive definiteness of a random matrix [duplicate]

Given an $n \times n$ symmetric random matrix whose entries have distribution $N(0,1)$, how to calculate the probability of positive definiteness of this matrix?
Mark Tuliy's user avatar
2 votes
0 answers
102 views

Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries

I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
Ian Cero's user avatar
  • 121
3 votes
0 answers
435 views

Rank of Hadamard product with random matrices

I do research in statistics and am not sure whether the following is considered research level or not in mathematics. If it isn't, I'm happy because that means the answer is probably known and I can ...
KOE's user avatar
  • 131
2 votes
0 answers
322 views

Expectation of square root of positive definite matrix

Let $U$ be a random matrix, supported on the positive-definite cone of matrices. We denote $\sqrt{U}$ to be the principal square root of $U$. That is, the unique positive-definite matrix such that $\...
Cain's user avatar
  • 393
4 votes
1 answer
294 views

Finding high-dimensional correlation matrices that are both sparse and low-rank

Let $\boldsymbol{R}$ be the correlation matrix of $X_i,i=1,\dots,p$ with a large $p\gg q=\text{rank}(\boldsymbol{R})$. Is that reasonable to assume that $\boldsymbol{R}$ is both (approximately) sparse ...
John's user avatar
  • 193
4 votes
1 answer
909 views

Uniformly Random Independent Unit Vectors Inner Product Limit

Suppose $V$ is a $N\times n$ matrix the columns of which are independently distributed uniformly on $\mathbf S^{N-1}$ the surface of the unit sphere in $\mathbf R^N$. I conjecture that $V^TV$ ...
Hans's user avatar
  • 2,239
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 ...
user avatar
13 votes
2 answers
879 views

The expected square of the determinant of a random row stochastic matrix

In this question Anthony Quas asks about the expected absolute value of the determinant of an $n\times n$ row stochastic matrix $A$, where the rows are independently selected from the uniform ...
Richard Stanley's user avatar
5 votes
1 answer
331 views

Matrix concentration bound

Suppose we have $N$ constant matrices $A_i \in R^{m\times m}, 1\leq i \leq N$. Consider $N$ random rotation-matrices $R_i \in SO(m), 1\leq i \leq N$. Is it possible to obtain a concentration bound on $...
Peter Huang's user avatar
1 vote
0 answers
167 views

Expected amount of linearly dependent random vectors? [closed]

Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$? EDIT: As said in the comments, I'm looking for the ...
Memphisd's user avatar
  • 123
5 votes
0 answers
133 views

Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$

Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
user173856's user avatar
  • 1,997
6 votes
3 answers
425 views

Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
Simd's user avatar
  • 3,377
5 votes
0 answers
327 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
user1189053's user avatar
-2 votes
3 answers
447 views

Determinant of matrix from set {-1, 1} [closed]

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$? I have ...
noone's user avatar
  • 45
27 votes
3 answers
13k views

What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
Andrew's user avatar
  • 433
2 votes
1 answer
2k views

Bounds on the eigenvalues of a random binary matrix

Consider $A$, a random binary matrix of zeros and ones in $\mathbb{R}^{{M\times N}}$, and $M>N$. We assume that $P(a_{i,j}=0)=P(a_{i,j}=1)=0.5$ (although I appreciate any advice on the case of non-...
Ali's user avatar
  • 127
2 votes
0 answers
458 views

Random variable matrix exponential

I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here. What ...
Winton's user avatar
  • 21
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 ...
Gil Kalai's user avatar
  • 24.7k
10 votes
1 answer
441 views

Probability that a random distance function is metric

Take a random $n \times n$ nonnegative symmetric matrix $D$ with zero diagonal. What is the probability that it is an abstract distance matrix, i.e. satisfies $D_{xy}+D_{yz} \geq D_{xz}$ for all index ...
Felix Goldberg's user avatar
42 votes
3 answers
5k views

The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
Denis Serre's user avatar
  • 52.3k
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 ...
Gil Kalai's user avatar
  • 24.7k
1 vote
1 answer
752 views

Random sampling a symmetric matrix

I'd like to sample the elements of a symmetric square matrix uniformly. For example, for a $N\times N$ matrix, I'd like to only keep $\alpha$% of the matrix elements to build a sparse matrix, while ...
WhitAngl's user avatar
  • 481
14 votes
1 answer
1k views

A Question on Random Matrices

Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by $$ V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q}) $$ where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
ght's user avatar
  • 3,626