For questions concerning random matrics.

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12 views

Distance of random span to a vector

I've been batteling with the following problem: Assume we have a diagonal matrix $D \in \mathbb{R}^{l \times l}$, a vector $\beta \in \mathbb{R}^l$. Next we simulate a random matrix (Idea inspired by ...
2
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0answers
13 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
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0answers
28 views

Products of Random Matrices

I'm interested in the following process on the space of $d \times d$ real valued matrices, $M_d(\mathbb{R})$. Fix $n \in \mathbb{N}$ and consider the process $$X_{k,n} = \left( I + ...
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12 views

a question which is somhow related to law of large number

suppose that p = [p1, p2, ..., pn]' is a random vector. (' == transpose) and each element of p like pi is a Gaussian random variable with zero mean (E(pi)=0) and variance vi (E(pi^2)=vi). the ...
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0answers
23 views

Expectation of the “second eigenvalue” of a random binary matrix

For a real matrix $A$, define its second eigenvalue to be $\max_{i\ge 2}|\lambda_i|$, where $\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_n$ are the eigenvalues of $A$. What is the expectation of the ...
4
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206 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
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0answers
12 views

Expected number of blocks in a polynomial of random 0-1 matrices

Fix $n\ge 1$ (integer) and $p\in[0,1]$. Let $A=A_{n,p}$ to be a random symmetric $n\times n$ 0-1 matrix, for which for each $1\le j<i\le n$, $\{A\}_{i,j}$ is 1 with probability $p$ and 0 otherwise, ...
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0answers
22 views

Maximal area of a monochromatic combinatorial rectangle

I'm stuck on this one and would appreciate any help: Let $M$ be a $2^n \times 2^n$random matrix of $0$'s and $1$'s (for each entry, there is probability of $\frac{1}{2}$ that it will be $0$, same ...
3
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80 views

What is the dual matrix (of a sample covariance matrix)?

Let $A$ be a matrix. I am most interested in the real, symmetric case, but for full understanding let's let $A$ be complex. What does it mean for $A^D$ to be the dual matrix of $A$? Can we interpret ...
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1answer
59 views

Random operators

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
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22 views

Density of Gaussian Unitary Ensemble

I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse. Take the Gaussian unitary ensemble (GUE) of $n ...
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10 views

Expected products of entries in a random unistochastic matrix

Consider the "unistochastic" distribution over $m \times m$ doubly-stochastic matrices obtained by taking the norm-squared of entries of a Haar-random unitary matrix. I'd like to know the expected ...
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0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
4
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2answers
70 views

Eigenvalues of a Random Matrix

I am studying the theory of random matrices lately, but there is a basic issue troubling my life. I hope someone here explain me this, thank you. A random matrix is defined as a matrix whose entries ...
3
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1answer
82 views

Random Matrix Theory and ESD

I need some help to understand what professor Terence Tao means in this part of "Topics in random matrix theory". I'm having a hard time to undertand this function ESD. How is possible for it to be ...
3
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0answers
19 views

Entries of a Haar distributed unitary matrix

The eigenvector matrix of a Wishart matrix is Haar distributed and that implies that the eigenvectors are uniformly distributed on a sphere. I'm interested to know what is the distribution of ...
2
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0answers
29 views

Eigenvalue distribution of sum of random matrices

Suppose we have a set of freely independent random matrices $X_i$ $i=\{1,\dots,n\}$ of dimension $M\times N$. Each $X_i$ has independent and identically distributed entries. Then we make the sum ...
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19 views

Are these two matrices asymptotically free?

The entries of $\mathbf{H}_1\in\mathbb{C}^{K\times M},\mathbf{H}_2\in\mathbb{C}^{K\times M},K<M$ are all i.i.d. RVs $\sim\mathcal{CN}(0,1)$. $\mathbf{D}\in\mathbb{R}^{K\times K}$ is a deterministic ...
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0answers
54 views

correlation matrix of the random matrix

$\mathbf{X}$ is random matrix given by $\mathbf{X}=\left[\begin{array}{*{20}{c}} \mathbf{x}_{1}&{}&{}&{}\\ {}&\mathbf{x}_{2}&{}&{}\\ {}&{}& \ddots &{}\\ ...
2
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1answer
61 views

What is the eigenvalue of matrix from matrix minus all-ones matrix?

Suppose we know the eigenvalues of matrix A, and J is all-ones matrix with all elements are one. Then what are the eigenvalues of A-J? ps. A is random matrix with element from distribution N(0,s) ...
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48 views

Problem about the determinant of a random matrix

I am being haunted about this problem on the value of the determinent of this Random Matrix ever since it came into my mind last week.The problem goes like this: Suppose $A$ is a square matrix of ...
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3answers
85 views

How sum work, vectors and matrices

I am having troubles understanding what is going on here. Would anyone be able to do this step by step with values so that I will be able to understand the SUM and how it works. xi is in this case ...
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0answers
32 views

Concentration Inequalities For Matrices? (Around Mean)

I need a result which talks about concentration of a 'random matrix' around its expected value. I need the following: $Pr(||X-E[X]|| \le \epsilon) \ge \hspace{2pt} ? \hspace{8pt}, X \in \mathbb{R}^{n ...
2
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2answers
76 views

Find the Norm of Matrix using Cauchy-Schwarz inequality

Let $A$ be $n \times n$ matrix and such that all of its entries are uniformly $O(1)$. Using Cauchy-Schwartz inequality, show that the operator norm of matrix $A$, which is $\|A\|_{op}=\sup_{x\in R^n: ...
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1answer
54 views

Relation between the eigenvalue density and the resolvent

Many texts (e.g. 1-2-3) on random matrices start with some variation of the identity: $$\rho_1(\lambda) = \frac{1}{\pi} \text{Im}\{\langle\text{Tr}(\mathbf{X}-\lambda\mathbf{I})^{-1}\rangle\}$$ ...
5
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1answer
95 views

Obtaining the Airy kernel from the Christoffel-Darboux formula with asymptotic Hermite polynomials

Let the Kernel associated to a family of orthogonal polynomial $p_n(x)$ with weight $w(x)$ be defined as $$K_N(x,y):=\frac{\sqrt{w(x)w(y)}}{\int w(x) p_{N-1}(x)p_{N-1}(x)dx} ...
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0answers
81 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
2
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0answers
58 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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0answers
32 views

The differentiation of the trace of complex matrix

Condition: all the matrices are complex. $\dagger$ denotes the conjugate transpose, $*$ denotes the conjugate, $\mathop{Trace}$ denote the trace of a matrix. What is the differentiations of the ...
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1answer
57 views

Calculating the Lyapunov index of elements of a matrix.

In Random Dynamical Systems - Arnold Ludwig, Example 3.3.9. The cocycle generated by a random matrix: $$A=\left(\begin{array}{cc} a & c\\ 0 & b \end{array}\right)$$ is given by: ...
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1answer
71 views

If a sequence of random matrices converge in probability, do their elements also converge?

Is it true that if a sequence of random matrices $\{X_n\}$ converge in probability to a random matrix $X_n\overset{P}{\to}X$ as $n\to\infty$ that the elements $X_n^{(i,j)}\overset{P}{\to} X^{(i,j)}$ ...
2
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0answers
89 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
0
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1answer
49 views

Change of variable formula, hermitian matrices

Let \begin{align} (d\mathbf{H})= \bigwedge_{1\leq j\leq k\leq N} d h_{jj}^{(1)} \bigwedge_{1\leq j< k\leq N}d h_{jk}^{(2)} \ ... \bigwedge_{1\leq j< k\leq N}\ d h_{jk}^{(\beta)} \end{align} ...
6
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1answer
97 views

The use of log in the Mean density of the nontrivial zeros of the Riemann zeta function (part 2)

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
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1answer
38 views

Meaning of symbol

In Furstenberg-Kesten theorem, a theory relating to products of random matrices, one of the assumptions is that: $$log^{+}||A||\in L^1(\mathbb{P}),$$ where $A$ (a random matrix) is the generator of ...
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1answer
40 views

Operators that are not represented as matrices , operating on matrices.

I am currently going through "Log-gases and random matrices" by PJ Forrester. I'm coming from a totally different academic background, and I cannot understand a point of his notation. More precisely, ...
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1answer
135 views

Jensen's inequality for frobenius norm

When I was going through a proof, I saw the following step: $$\frac{1}{p}\operatorname{trace}(\Sigma) \leq \frac{1}{\sqrt{p}} \|\Sigma\|$$ where, $p$ is the number of variables, $\sigma$ is the ...
0
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1answer
23 views

Unitary matrices and Riemann zeros is there an error?

when they exploit the relationship by Berry and Keating between the Riemann zeros and eigenvalues of random matrices why do they choose $$ \frac{\gamma _{n}}{2\pi}log \frac{\gamma}{2\pi e} $$ as a ...
2
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1answer
44 views

Question regard the notion of almost sure convergence

Consider an $n\times m$ matrix with i.i.d. entries each having zero mean and variance $1/n$. Let $Y = X^TX$. By the strong law of large numbers, we know that the $(i,j)$ entry of $Y$ goes almost ...
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0answers
29 views

Variance of spectral norm of random matrix

Given $A$, a $n\times n$ random matrix with centred Gaussian (real) i.i.d entries with variance $\sigma$, what is the variance in the spectral norm $\left(\sqrt{\text{Largest eigenvalue of } A^\dagger ...
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0answers
16 views

where should I start if I want to understand the relationship between sample size and eigenvalues?

Here is my problem: I have a sample size $N$ for d-dimension vector $v$, and I calculated out a set of eigenvalues $T1$ for the sample covariance for it. I want to know, if I replaced $n$ samples out ...
2
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1answer
57 views

Random Matrix Question

Let $A$ be a $n\times m$ ($n<m$) random matrix with normal i.i.d. entries ($N(0,1)$). Using the law of large numbers, it readily can be shown that as $n\to\infty$ $$ \frac{1}{n}A^TA\to I_m $$ ...
3
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0answers
64 views

trace class norms of random matrices

We denote by $||.||_1$ the trace class norm. on $M_n$.Let $(r_{ij})_{1 \leq i,j\leq n}$ be a family of independent identically distributed random variables which take the values $-1$ and $1$ with ...
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0answers
39 views

near- rank - deficiency property

I am trying to understand low rank approximations and when I was reading up on it, it is stated that one of the methods of getting the low rank approximation is the near-rank-deficiency property holds ...
2
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1answer
35 views

A simple question regarding $df(E)$ for $f(A)=A^{-1}$

I am trying to follow the document on http://web.mit.edu/people/raj/Acta05rmt.pdf, and I got a simple question that why for $f(A)=A^{-1}$ we have $df(E)=-A^{-1}EA^{-1}$ on page 5? (where $E$ is a ...
5
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2answers
227 views

Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent

Let's take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$) (of course the problem is ...
1
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1answer
137 views

How to generate a rectangular matrix that has no left/right inverse?

Is it possible to generate non-invertible rectangular matrices? If so, how we can prove it? And is there any way to generate randomly such matrices in MATLAB (with preferably uniform distribution)?
3
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0answers
30 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
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2answers
112 views

the product of a matrix and a permutation matrix

Can a permutation matrix ($P$) be used to change the rank of another matrix ($M$)? Is there any literature to this effect, or to the contrary? I've tried a few small examples and the resulting matrix ...
5
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2answers
277 views

Random binary matrix with given rows and columns sums

I need to generate a random binary matrix $(n, n)$ whose rows sums and columns sums are $4$. I don't manage to find a quite efficient algorithm to do this. Have you an idea please ? NB : The ...